CHEMPHYSCHEM ARTICLES DOI: 10.1002/cphc.201301048
Molecular Theory of Phase Separation in Nematic Liquid Crystals Doped with Spherical Nanoparticles Mikhail A. Osipov*[a] and Maxim V. Gorkunov[b] A molecular-statistical theory is developed, which enables one to describe the nematic–isotropic phase transition in liquid crystals doped with spherical nanoparticles taking into account the effects of phase separation. It has been shown that in the case of strong interaction between nanoparticles and mesogenic molecules the nematic nanocomposite possesses a number of unexpected properties. In particular, the nematic– isotropic co-existence region appears to be very broad, and the system either undergoes a direct transition from the isotropic phase into the phase-separated state, or undergoes the
transition into the homogeneous nematic phase first and then phase-separates at a lower temperature. Phase separation does not occur at all if the concentration of the nanoparticles is sufficiently low, and in some cases it takes place only within a finite region of nanoparticle concentration. A number of temperature–concentration phase diagrams is presented and the molar fractions of nanoparticles in the co-existing isotropic and nematic phases are calculated numerically as functions of temperature.
1. Introduction Nematic liquid crystals (LCs) doped with nanoparticles (NPs) are considered to be very promising novel materials for a broad range of applications. It has been shown by various authors that doping of a nematic LC with a small amount of metal, dielectric, or semiconductor NPs affects a number of important properties of nematic materials. In particular, this results in a decrease of switching voltages and reduction of the switching times of LC displays (see, for example, refs. [1–5]). Also, nematics doped with ferroelectric NPs are characterized by an enhanced dielectric and optical anisotropy, increased electro-optic response,[7, 8] and improved photorefractive properties.[9] It has also been shown both experimentally and theoretically that the properties of the nematic–isotropic (N-I) phase transition may be significantly affected by the presence of NPs. For example, a decrease of the N-I transition temperature is observed in nematics doped with approximately isotropic silver,[10] gold,[11] or aerosil particles,[12, 13] while the N-I transition temperature increases if the nematic LC is doped with strongly anisotropic NPs including nanotubes,[14] magnetic nanorods,[15] and ferroelectric particles.[16, 17] Recently a mean-field molecular theory has been developed,[18, 19] which describes these effects. In the case of spherical or weakly anisotropic NPs, the N-I transition temperature decreases with the decreasing concentration of NPs and as a result the nematic phase is partially de[a] Prof. M. A. Osipov Department of Mathematics, University of Strathclyde 26 Richmond Str., Glasgow G1 1XH (UK) E-mail: [email protected] [b] Dr. M. V. Gorkunov A. V. Shubnikov Institute of Crystallography Russian Academy of Sciences 119333 Moscow (Russia)
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stabilized. The total free energy of such a system may be minimized if it separates into the isotropic phase with an increased NPconcentration and the nematic phase with a lower NPconcentration. One notes that such a phase separation is much different from an ordinary demixing, which takes place already in the isotropic phase and does not require the system to undergo a phase transition. Experimentally such a demixing can be suppressed by attaching appropriate organic groups to the surface of the NPs which makes them more compatible with the surrounding fluid. In contrast, the origin of the phase separation is closely related to the phase-transition thermodynamics, which we find interesting from both the fundamental and the applications point of view, and it has neither been studied experimentally nor theoretically so far. It should be noted that a similar phase separation occurs around the N-I transition point in mixtures of different LCs and, in particular, in nematics doped with nonmesogenic molecules (see for example, refs. [20, 21]). The corresponding twophase region around the N-I transition, however, is usually very narrow. This is related to the fact that properties of the dopant molecules do not differ much from those of the host ones. In contrast, the properties of metal or semiconductor NPs may differ very significantly from those of typical mesogenic molecules, and, as a result, the region of co-existence of the isotropic and the nematic phase may be much wider.[22] In this paper we develop a mean-field molecular theory of the phase separation in nematic LCs doped with spherical NPs and show that, depending on the relative strength of interaction between NPs and mesogenic molecules, the nematic and the isotropic phases may co-exist over a broad temperature interval. A number of phase diagrams were obtained, which indicate that the phase separation usually occurs above a certain threshold value of the NP molar fraction. Moreover, in some ChemPhysChem 0000, 00, 1 – 7
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cases the phase separation only occurs within a finite window of NP concentration and inside the nematic phase.
where S is the nematic order parameter of the mesogenic molecules, which is expressed as Equation (5):
2. Molecular Theory
S¼
Let us consider a nematic LC doped with a small amount of spherical NPs. In the molecular theory one has to take into account both isotropic and anisotropic interactions between mesogenic groups, isotropic interactions between mesogenic groups and spherical NPs, and also isotropic interactions between the NPs themselves. Then the system can be characterized by the following total intermolecular interaction potential averaged over all intermolecular vectors [Eq. (1)]: H¼
1 X Vmm ðai aj Þ þ Unn þ Unm þ Umm 2 ij
ð2Þ
where fm ðaÞ is the orientational distribution function of the mesogenic molecules, and 1m and 1n are the number densities of the mesogenic molecules and the NPs correspondingly. The function Vmm ðai aj Þ can be expanded in Legendre polynomials Pn ðai aj Þ taking into account the first nonpolar term which is responsible for the nematic ordering [Eq. (3)]: ð3Þ
Here P2 ðxÞ ¼ 3x 2 =2 1=3 is the second Legendre polynomial. Substituting this expansion into Equation (2) one obtains the following Maier–Saupe-type free energy of the nematic composite: 1 F ¼kB T1n ðln1n 1Þ þ kB T1m ðln1m 1Þ V N 1 1 12n Unn 12m Umm þ 1m 1n Umn 2 2 Z 1 2 2 þ 1m JS kB T fm ðaÞlnfm ðaÞda 2 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
ð5Þ
where n is the nematic director. Minimizing the free energy [Eq. (4)] with respect to the orientational distribution function fm ðaÞ and substituting the equilibrium expression for fm ðaÞ back into Equation (4) one obtains Equation (6): 1 F ¼kB T1n ðln1n 1Þ þ kB T1m ðln1m 1Þ V N 1 1 1 12n Unn 12m Umm þ 1m 1n Umn 12m JS2 kB TlnZ 2 2 2
ð6Þ
where [Eq. (7)]:
In the mean-field approximation, the free energy of the nematic phase can be written in the form of Equation (2) (see, e.g. ref. [23]):
Vmm ðai aj Þ U0 þ JP2 ðai aj Þ
P2 ða nÞfm ðaÞda
ð1Þ
where Unn ; Umm , and Unm are the average isotropic interaction potentials between the NPs, mesogenic molecules, and a NP and a mesogenic group, respectively. Vmm ðai aj Þ is the anisotropic interaction potential between the mesogenic molecules, which depends on the unit vectors ai and aj in the direction of the primary axis of the molecules i and j, respectively.
1 F ¼kB T1n ðln1n 1Þ þ kB T1m ðln1m 1Þ V N 1 2 1 1 U 12 U þ 1m 1n Umn 2 n nn 2 m mm Z 1 þ 12m Vmm ðai aj Þfm ðai Þfm ðaj Þdai daj 2 Z þkB T fm ðaÞlnfm ðaÞda
Z
ð4Þ
ZN ¼
Z
p
exp½b1m JSP2 ðcosgÞsingdg
ð7Þ
0
and where the nematic order parameter S satisfies the self-consistent Equation (8): S¼
1 ZN
Z
p
P2 ðcosgÞexp½b1m JSP2 ðcosgÞsingdg
ð8Þ
0
The free energy of the isotropic phase is obtained by setting S ¼ 0 [Eq. (9)]: 1 F ¼kB T1n ðln1n 1Þ þ kB T1m ðln1m 1Þ V I 1 1 12n Unn 12m Umm þ 1m 1n Umn 2 2
ð9Þ
3. Phase Separation One notes that the concentration of the NPs in the nematic phase is generally different from that in the co-existing isotropic phase, and may it be strongly temperature dependent. The co-existence between the nematic and the isotropic phases in the system under consideration is possible only if the chemical potentials of both the NPs and the mesogenic molecules are the same in the two phases. The pressure must also be the same in the two phases. However, for incompressible LCs only the equations for the chemical potentials are relevant, that is, mnI ¼ mnN and mmI ¼ mmN , where mnI and mnN are the chemical potentials of the NPs in the isotropic and in the nematic phase, respectively; and mmI and mmN are the corresponding chemical potentials of the mesogenic molecules. Using the well-known general equation for the chemical potential one obtains the following system [Eq. (10)] of two simultaneous equations: 1 @FI 1 @FN 1 @FI 1 @FN ¼ , ¼ VI @1n VN @1n VI @1m VN @1m ChemPhysChem 0000, 00, 1 – 7
ð10Þ
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Substituting Equations (9) and (6) for the free energies of the isotropic and the nematic phase into Equation (10) one obtains the Equation (11): 1 ln mN ¼ U1 ð1mN 1mI Þ þ U12 ð1nN 1nB Þ þ lnZN 1mI
1nN ¼ U2 ð1nN 1nI Þ þ U12 ð1mN 1mI Þ 1nI
ð12Þ
Neglecting a small density change at the transition, the number densities of both the NPs and the mesogenic groups in the nematic and in the isotropic phase can be expressed in terms of the volume fraction i of the NPs in the corresponding phase i [Eq. (13)]: ð13Þ
where i ¼ N; I, 1m0 is approximately equal to the number density of the mesogenic groups in the pure LC, and 1n0 can be estimated as 1n0 :1=vn , where vn is the NP volume. Now Equations (11) and (12) can be expressed in terms of the two variables N and I [Eqs. (14, 15)]: ln
1 N ¼ w1 ðI N Þ þ lnZN 1 I
ð14Þ
ln
I ¼ w2 ðI N Þ N
ð15Þ
where [Eq. (16)]: w1 ¼ ð1m0 U1 1n0 U12 Þ, w2 ¼ ð1n0 U2 1m0 U12 Þ
ð16Þ
One notes that Equations (14) and (15) contain only two constants, w1 ; w2 , and it can readily be shown that the phase co-existence is possible only if w2 > 1. If the isotropic attraction between the NPs and the mesogenic molecules is much stronger than that between the mesogenic molecules, the inequality 1n0 U12 > 1m0 U1 is satisfied, and hence w1 < 0. The numerical solution of Equations (14), (15) together with (7) and the self-consistent Equation (8) for the nematic order parameter is significantly simplified if the volume fraction of the NPs is sufficiently small in both phases, that is, I ¼1 and N ¼1. In this case Equation (14) is simplified as [Eq. (17)]: ð1 w1 ÞðI N Þ ¼ lnZN
ð17Þ
One notes also that the partition function ZN is independent of I , and therefore I can be excluded from the system of simultaneous equations for I ¼1 and N ¼1 which results in 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
lnZN N ð1 w1 Þ
ð18Þ
Now N can be found by solving Equation (18) numerically, and I can then be evaluated in terms of N as Equation (19):
where we have introduced the non-dimensional interaction eff constants U1 ¼ Umm =ðkTB Þ, U2 ¼ Ueff nn =ðkTB Þ, U12 ¼ Unm =ðkTB Þ, and the number densities of the mesogenic molecules (1mI and 1mN ) and the NPs (1nI and 1nN ) in the isotropic and nematic phases correspondingly.
1ni ¼ 1n0 i , 1mi ¼ 1m0 ð1 i Þ
ZNw2 =ð1w1 Þ ¼ 1 þ
ð11Þ
and Equation (12) ln
a single equation for N [Eq. (18)]:
I ¼ N þ
lnZN 1 w1
ð19Þ
One notes that the volume fractions of the NPs in the nematic and the isotropic phases are not completely independent. Indeed, in the experiment one normally controls the total number of the NPs, Nn , in the volume V which yields the average volume fraction of the NPs, . It follows from the conservation of the total number of the NPs that V ¼ I VI þ N VN , where V ¼ VI þ VN is the total volume of the system. On the other hand, solutions of Equations (14) and (15) are independent of . From these equations one obtains the following expressions for the volumes VN and VI [Eq. (20)]: VI ¼ V
I , V ¼V N I N I N N
ð20Þ
One can readily see from Equation (20) that the phase co-existence is possible only if N < < I as N < I . Taking into account that N and I are independent of , one concludes that if is outside the interval ðN ; I Þ, the phase co-existence is impossible and only one phase may be stable at a given temperature. It should also be taken into account that the co-existing nematic and isotropic phases are globally stable only if the total free energy of the phase-separated system is lower than the free energy of both the isotropic and the nematic homogeneous phases. This condition can be expressed as Equation (21): sep FNI Fhom 1 VI VN ¼ F ð Þ þ FN ðN Þ FI;N ðÞ < 0 VkB T V I I VkB T V
ð21Þ
where VI and VN are given by Equation (20) and the free energy densities FI ðÞ and FN ðÞ can be expressed as Equations (22) and (23) using Equations (6), (9), and (13): FI ðÞ=VkB T ¼1n0 ln þ 1m0 ð1 Þlnð1 Þ 1 1 12n0 2 U2 12m0 ð1 Þ2 U1 þ 1m0 1n0 ð1 ÞU12 2 2 ð22Þ FN ðÞ=VkB T ¼FI ðÞ=VkB T 1 12m0 ð1 Þ2 J* S2m 1m0 ð1 ÞlnZN 2
ð23Þ
where J* ¼ J=kB T. Note that for the separated state in Equation (21) the nematic order parameter is SðN Þ in the nematic phase, which co-exists with the isotropic one, and is given by ChemPhysChem 0000, 00, 1 – 7
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CHEMPHYSCHEM ARTICLES Equation (8) with 1m ¼ 1m0 ð1 N Þ and ZN ¼ ZN ðN Þ. At the same time, the order parameter of the homogeneous nematic phase SðÞ is given by the same equation with 1m ¼ 1m0 ð1 Þ and ZN ¼ ZN ðÞ. Note that evaluation of Equation (21) can be considerably simplified if the volume of a NP is significantly larger than that of a mesogenic group, and therefore 1n =1m ¼1.
4. Phase Diagrams with Co-Existing Nematic and Isotropic Phases Let us first consider Equations (14) and (15), which enable one to determine the molar fractions of the NPs in co-existing phases as functions of temperature. A typical numerical solution of these equations for moderately strong interactions between the NPs and the mesogenic molecules is presented in Figure 1 One notes that there exists a bifurcation point which
www.chemphyschem.org system phase separates, because the separated state is globally stable only if the total free energy of the separated system is lower than that of any homogeneous phase at the same temperature and the same NP concentration [see Eq. (21)]. One notes also that if the value of is too small or too large (see the dotted line ¼ 0:01 in Figure 1), there may be no intersection at all, that is the condition N < < I is not satisfied at any temperature and, therefore, the system never phase-separates. The solutions for N and I together with Equation (21) were used to compose the temperature–concentration phase diagram presented in Figure 2 for the same values of the interac-
Figure 2. Phase diagram of the composite calculated from the mean-field theory with interaction constants w1 ¼ 5 and w2 ¼ 10.
Figure 1. NP volume fractions in co-existing isotropic (upper curve) and nematic (lower curve) phases of the composites with w1 ¼ 5 and w2 ¼ 10.
corresponds to a critical temperature Tc . At higher temperatures there is no solution, that is, the phase separation is impossible, and directly below the critical temperature the difference between the molar fractions of the NPs in the two phases is fairly small. It should be noted, however, that in the general case only part of this solution corresponds to an actual physical state of the system. As discussed in the previous section, Equations (14) and (15) do not depend on the average molar fraction of the NPs , which can be controlled experimentally. At the same time, the value of must lie between the two branches in Figure 1, as N < < N . If is different from the critical value of c at the bifurcation point, the system separates into the nematic and the isotropic phases with a finite difference between the NP molar fractions and finite volumes of the two phases. In this general case the separation occurs at some temperature Tsep which is below the bifurcation temperature Tc , and which is an intersection of the horizontal line and one of the curves representing N ðTÞ or I ðTÞ (see the intersection of N ðTÞ and the horizontal dashed line ¼ 0:04 in Figure 1). In principle, even if T < Tsep one cannot conclude that the 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
tion constants w1 and w2 as in Figure 1. One notes that there is no phase separation at sufficiently low concentration of the NPs. In this domain, the N–I transition temperature decreases with increasing owing to the dilution effect, which has been considered in detail in ref. [18]. Above a certain critical concentration, the N–T phase transition is accompanied by the separation between the isotropic and the nematic phase, and the two phases co-exist over a significant temperature interval. In this region the transition temperature of the phase-separated state decreases more slowly than the N–I transition temperature at low concentrations. In this case, another critical value of the NP molar fraction shows up as f * , shown in Figure 2. If f > f * , the system undergoes a direct transition from the isotropic into the the phase-separated state, in which the isotropic and the nematic phases co-exist. In contrast, if f < f * , the system first undergoes a transition into the homogeneous nematic phase and then phase-separates at a somewhat lower temperature. The solution of Equations (14) and (15) and the corresponding phase diagrams for a nematic composite with very strong interaction between the NPs and the mesogenic molecules are presented in Figures 3 and 4, respectively. In this case the temperature range of the co-existence between the nematic and the isotropic phase is more narrow, and there is only a relatively small NP concentration window, in which the separation takes place at all. One notes that in this case a reentrant homogeneChemPhysChem 0000, 00, 1 – 7
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5. Discussion
Figure 3. The same as in Figure 1 with w1 ¼ 10 and w2 ¼ 30.
Figure 4. The same as in Figure 2 with w1 ¼ 10 and w2 ¼ 30.
ous nematic phase may occur within a narrow interval of the NP molar fraction. In contrast, for weaker interactions between the NPs and the mesogenic molecules (smaller absolute values of w1 ), the region of phase-separated state expands considerably (compare Figures 2 and 5). Decreasing absolute values of jw1 j result in a proportionate decrease of the critical total concentration * . Additionally, in this case the phase-separated state dominates over the low-temperature part of the phase diagram and even at small total NP concentrations the homogeneous
Figure 5. The same as in Figure 2 with w1 ¼ 1 and w2 ¼ 10.
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We have developed a mean-field molecular theory of nematic LCs doped with spherical NPs taking into account a possibility of phase separation between the nematic and the isotropic phases and considering strong isotropic interactions between the NPs and the mesogenic molecules The results appear to be rather unusual and in any case different from what one may expect from the behavior of mixtures of nematics with small non-mesogenic dopants. In particular, the temperature range of the co-existing nematic and isotropic phases may be very large. It is well known that in mixtures of LCs the N–I coexistence region is typically of the order of 1–2 degrees and can often be neglected. In contrast, the theory indicates that in nematic nano-composites the co-existence region may be as broad as the nematic phase itself. Secondly, the phase separation does not occur at sufficiently low NP concentrations because the mixing entropy is dominating the behavior of the system. Moreover, within a certain range of the NP concentration the composite may undergo a direct transition from the isotropic phase into the phase-separated state, while at other NP concentrations the system first undergoes a transition into the homogeneous nematic phase and then into the phase-separated state at lower temperature. In nano-composites with very strong interactions between the NPs themselves and between the NPs and the mesogenic molecules, the N–I phase separation may only occur within a finite NP concentration interval, and a reentrant homogeneous nematic phase may be stable for some concentrations. These unexpected results are mainly related to the fact that the properties of the NPs differ very much from those of typical mesogenic molecules. In particular, the isotropic interactions between the NPs and the mesogenic groups (and between the NPs themselves) is expected to be significantly stronger than those between the mesogenic molecules. This is mainly due to the large effective volume of a typical spherical NP, which also includes organic chains attached to the surface of the metal or semiconductor core. As discussed in detail in this paper, the total free energy may be minimized if the system separates into the isotropic phase with an increased NP concentration and the nematic phase with a lower NP concentration. This kind of phase separation has been observed in few anisotropic soft-matter systems.[2, 6] However, to the best of our knowledge, it has not been studied in detail experimentally so far. Very recently, however, a two-step decrease of both the N–I transition temperature and the transition heat in few polymer and low molecular weight LCs doped with quantum dots[22] has been interpreted assuming that the system separated in to the isotropic and the nematic phase with different concentrations of quantum dots. A different type of phase separation has recently been observed in refs. [27, 28]. It has been shown experimentally that gold NPs with mesogenic coatings form reversible networks composed of nematic droplets accompanied by disclination ChemPhysChem 0000, 00, 1 – 7
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CHEMPHYSCHEM ARTICLES lines and loops as a result of a specific phase separation along with the enrichment of NPs at the N–I liquid interfaces. In addition, the properties of nematic LCs doped with metal, dielectric, or semiconductor NPs differ significantly from those of nematics with relatively large colloidal particles. For example, it has been shown theoretically in ref. [29] that in nematics doped with colloidal particles phase separation at the N–I transition occurs at all particle concentrations, that is, there has been no threshold concentration found. This has been explained by the large size of the colloidal particles, which produce director distortions in the surrounding LC medium. Such director distortions have been modeled in ref. [29] using a random anisotropic field which results in a different mathematical form of the free energy of the mixture. As a result the tendency for the separation at low concentrations of colloidal particles is strongly enhanced. A very broad N–I co-existence region in LCs doped with colloidal particles has also been described in refs. [24, 25] using simple mean-field theory of mixtures. This theory, however, is based on a different (and rather oversimplified) model interaction potential that is more suitable for large colloidal particles. In our paper we have not taken into consideration possible formation of chains and other types of NP aggregation, which, in principle, may strongly modify the conditions for N–I phase separation. There exists some experimental evidence that quantum dots may form long chains in nematic LCs.[26] Large values of the dielectric anisotropy in the nematic phase doped with ferroelectric particles have also indicated that strongly polar particles may form chains.[16] Chain ordering of NPs thus definitely deserves a separate investigation.
Acknowledgements This work was supported by the Ministry of Education and Science of Russian Federation (agreement 8509) and the Russian Foundation for Basic Research (project No. 13-03-00579 a). Keywords: liquid crystals · molecular theory · nanoparticles · nematic phase · phase transition [1] H. Qi, B. Kinkead, T. Hegmann, Proc. SPIE Int. Soc. Opt. Eng. 2008, 6911, 691106.
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www.chemphyschem.org [2] H. Qi, T. Hegmann, J. Mater. Chem. 2006, 16, 4197. [3] Y. Shiraishi, N. Toshima, K. Maeds, H. Yoshikawa, J. Xu, S. Kobayashi, Appl. Phys. Lett. 2002, 81, 2845. [4] S. Kobayashi, N. Toshima, Inf. Disp. 2007, 23, 26 – 28. [5] H. Yoshida, K. Kawamoto, H. Kubo, T. Tsuda, A. Fujii, S. Kuwabata, M. Ozaki, Adv. Mater. 2010, 22, 622. [6] J. Yamamoto, H. Tanaka, Nature 2001, 409, 321. [7] S. Kaur, S. P. Singh, A. M. Biradar, A. Choudhary, K. Sreeniva, Appl. Phys. Lett. 2007, 91, 023120. [8] A. Kumar, J. Prakash, D. S. Mehta, A. M. Biradar, W. Haase, Appl. Phys. Lett. 2009, 95, 023117. [9] O. Buchnev, A. Dyadyusha, M. Kaczmarek, V. Reshetnyak, Yu. Reznikov, J. Opt. Soc. Am. B 2007, 24, 1512. [10] E. B. Barmatov, D. A. Pebalk, M. V. Barmatova, Liq. Cryst. 2006, 33, 1059. [11] P. Kopcˇansky´, N. Tomasˇovicˇov, M. Konerack, M. Timko, Z. Mitrov, V. Zvisˇov, N. ber, K. Fodor-Csorba, T. Tth-Katona, A. Vajda J. Jadzyn, E. Beaugnon, X. Chau, Acta Phys. Pol. A 2010, 118, 988. [12] G. Sinha, C. Glorieux, J. Thoen, Phys. Rev. E 2004, 69, 031707. [13] T. Bellini, M. Buscagli, C. Chiccoli, F. Mantegazza, P. Pasini, C. Zannoni, Phys. Rev. Lett. 2000, 85, 1008. [14] H. Duran, B. Gazdecki, A. Yamashita, T. Kyu, Liq. Cryst. 2005, 32, 815. [15] P. Kopcˇansky´, N. Tomasˇovicˇov, M. Konerack, M. Timko, V. Zvisˇov, A. Dzˇarov, J. Jadzyn, E. Beaugnon, X. Chaud, Int. J. Thermophys. 2011, 32, 807. [16] Yu. Reznikov, O. Buchnev, O. Tereshchenko, V. Reshetnyak, A. Glushchenko, J. West, Appl. Phys. Lett. 2003, 82, 1917. [17] F. Li, O. Buchnev, C. I. Cheon, A. Glushchenko, V. Reshetnyak, Yu. Reznikov, T. J. Sluckin, J. L. West, Phys. Rev. Lett. 2006, 97, 147801; F. Li, O. Buchnev, C. I. Cheon, A. Glushchenko, V. Reshetnyak, Yu. Reznikov, T. J. Sluckin, J. L. West, Phys. Rev. Lett. 2007, 99, 219901(E). [18] M. V. Gorkunov, M. A. Osipov, Soft Matter 2011, 7, 4348. [19] L. M. Lopatina, J. R. Selinger, Phys. Rev. E 2011, 84, 041703. [20] G. W. Gray, Molecular Structure and the Properties of Liquid Crystals, Academic Press, New York, 1962. [21] H. T. Peterson, D. E. Martire, Mol. Cryst. Liq. Cryst. 1974, 25, 89. [22] M. V. Gorkunov, G. A. Shandryuk, A. M. Shatalova, I. Yu. Kutergina, A. S. Merekalov, Ya. V. Kudryavtsev, R. V. Talroze, M. A. Osipov, Soft Matter 2013, 9, 3578. [23] M. A. Osipov, Molecular Theories of Liquid Crystals in Handbook of Liquid Crystals, Vol. 1, 2nd ed. (Eds.: D. Demus, J. Goodby, G. W. Gray, H.-W. Spiess, V. Vill), Wiley-VCH, Weinheim, 1998. [24] A. Matsuyama, R. Hirashima, J. Chem. Phys. 2008, 128, 044907. [25] A. Matsuyama, J. Chem. Phys. 2009, 131, 204904. [26] R. Basu, G. S. Iannacchione, Phys. Rev. E 2009, 80, 010701(R). [27] J. Milette, S. J. Cowling, V. Toader, C. Lavigne, I. M. Saez, R. B. Lennox, J. W. Goodby, L. Reven, Soft Matter 2012, 8, 173. [28] J. Milette, S. Relaix, C. Lavigne, V. Toader, S. J. Cowling, I. M. Saez, R. B. Lennox, J. W. Goodby, L. Reven, Soft Matter 2012, 8, 6593. [29] V. Poa-Nita, P. van der Shoot, S. Ktalj, Eur. Phys. J. E 2006, 21, 189.
Received: November 11, 2013 Published online on && &&, 2014
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ARTICLES Transit: A molecular-statistical theory is developed which enables one to describe the nematic-isotropic phase transition in liquid crystals doped with spherical nanoparticles taking into account the effects of phase separation.
2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
M. A. Osipov,* M. V. Gorkunov && – && Molecular Theory of Phase Separation in Nematic Liquid Crystals Doped with Spherical Nanoparticles
ChemPhysChem 0000, 00, 1 – 7
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Molecular Theory of Phase Separation in Nematic Liquid Crystals Doped with Spherical Nanoparticles Mikhail A. Osipov*[a] and Maxim V. Gorkunov[b] A molecular-statistical theory is developed, which enables one to describe the nematic–isotropic phase transition in liquid crystals doped with spherical nanoparticles taking into account the effects of phase separation. It has been shown that in the case of strong interaction between nanoparticles and mesogenic molecules the nematic nanocomposite possesses a number of unexpected properties. In particular, the nematic– isotropic co-existence region appears to be very broad, and the system either undergoes a direct transition from the isotropic phase into the phase-separated state, or undergoes the
transition into the homogeneous nematic phase first and then phase-separates at a lower temperature. Phase separation does not occur at all if the concentration of the nanoparticles is sufficiently low, and in some cases it takes place only within a finite region of nanoparticle concentration. A number of temperature–concentration phase diagrams is presented and the molar fractions of nanoparticles in the co-existing isotropic and nematic phases are calculated numerically as functions of temperature.
1. Introduction Nematic liquid crystals (LCs) doped with nanoparticles (NPs) are considered to be very promising novel materials for a broad range of applications. It has been shown by various authors that doping of a nematic LC with a small amount of metal, dielectric, or semiconductor NPs affects a number of important properties of nematic materials. In particular, this results in a decrease of switching voltages and reduction of the switching times of LC displays (see, for example, refs. [1–5]). Also, nematics doped with ferroelectric NPs are characterized by an enhanced dielectric and optical anisotropy, increased electro-optic response,[7, 8] and improved photorefractive properties.[9] It has also been shown both experimentally and theoretically that the properties of the nematic–isotropic (N-I) phase transition may be significantly affected by the presence of NPs. For example, a decrease of the N-I transition temperature is observed in nematics doped with approximately isotropic silver,[10] gold,[11] or aerosil particles,[12, 13] while the N-I transition temperature increases if the nematic LC is doped with strongly anisotropic NPs including nanotubes,[14] magnetic nanorods,[15] and ferroelectric particles.[16, 17] Recently a mean-field molecular theory has been developed,[18, 19] which describes these effects. In the case of spherical or weakly anisotropic NPs, the N-I transition temperature decreases with the decreasing concentration of NPs and as a result the nematic phase is partially de[a] Prof. M. A. Osipov Department of Mathematics, University of Strathclyde 26 Richmond Str., Glasgow G1 1XH (UK) E-mail: [email protected] [b] Dr. M. V. Gorkunov A. V. Shubnikov Institute of Crystallography Russian Academy of Sciences 119333 Moscow (Russia)
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stabilized. The total free energy of such a system may be minimized if it separates into the isotropic phase with an increased NPconcentration and the nematic phase with a lower NPconcentration. One notes that such a phase separation is much different from an ordinary demixing, which takes place already in the isotropic phase and does not require the system to undergo a phase transition. Experimentally such a demixing can be suppressed by attaching appropriate organic groups to the surface of the NPs which makes them more compatible with the surrounding fluid. In contrast, the origin of the phase separation is closely related to the phase-transition thermodynamics, which we find interesting from both the fundamental and the applications point of view, and it has neither been studied experimentally nor theoretically so far. It should be noted that a similar phase separation occurs around the N-I transition point in mixtures of different LCs and, in particular, in nematics doped with nonmesogenic molecules (see for example, refs. [20, 21]). The corresponding twophase region around the N-I transition, however, is usually very narrow. This is related to the fact that properties of the dopant molecules do not differ much from those of the host ones. In contrast, the properties of metal or semiconductor NPs may differ very significantly from those of typical mesogenic molecules, and, as a result, the region of co-existence of the isotropic and the nematic phase may be much wider.[22] In this paper we develop a mean-field molecular theory of the phase separation in nematic LCs doped with spherical NPs and show that, depending on the relative strength of interaction between NPs and mesogenic molecules, the nematic and the isotropic phases may co-exist over a broad temperature interval. A number of phase diagrams were obtained, which indicate that the phase separation usually occurs above a certain threshold value of the NP molar fraction. Moreover, in some ChemPhysChem 0000, 00, 1 – 7
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cases the phase separation only occurs within a finite window of NP concentration and inside the nematic phase.
where S is the nematic order parameter of the mesogenic molecules, which is expressed as Equation (5):
2. Molecular Theory
S¼
Let us consider a nematic LC doped with a small amount of spherical NPs. In the molecular theory one has to take into account both isotropic and anisotropic interactions between mesogenic groups, isotropic interactions between mesogenic groups and spherical NPs, and also isotropic interactions between the NPs themselves. Then the system can be characterized by the following total intermolecular interaction potential averaged over all intermolecular vectors [Eq. (1)]: H¼
1 X Vmm ðai aj Þ þ Unn þ Unm þ Umm 2 ij
ð2Þ
where fm ðaÞ is the orientational distribution function of the mesogenic molecules, and 1m and 1n are the number densities of the mesogenic molecules and the NPs correspondingly. The function Vmm ðai aj Þ can be expanded in Legendre polynomials Pn ðai aj Þ taking into account the first nonpolar term which is responsible for the nematic ordering [Eq. (3)]: ð3Þ
Here P2 ðxÞ ¼ 3x 2 =2 1=3 is the second Legendre polynomial. Substituting this expansion into Equation (2) one obtains the following Maier–Saupe-type free energy of the nematic composite: 1 F ¼kB T1n ðln1n 1Þ þ kB T1m ðln1m 1Þ V N 1 1 12n Unn 12m Umm þ 1m 1n Umn 2 2 Z 1 2 2 þ 1m JS kB T fm ðaÞlnfm ðaÞda 2 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
ð5Þ
where n is the nematic director. Minimizing the free energy [Eq. (4)] with respect to the orientational distribution function fm ðaÞ and substituting the equilibrium expression for fm ðaÞ back into Equation (4) one obtains Equation (6): 1 F ¼kB T1n ðln1n 1Þ þ kB T1m ðln1m 1Þ V N 1 1 1 12n Unn 12m Umm þ 1m 1n Umn 12m JS2 kB TlnZ 2 2 2
ð6Þ
where [Eq. (7)]:
In the mean-field approximation, the free energy of the nematic phase can be written in the form of Equation (2) (see, e.g. ref. [23]):
Vmm ðai aj Þ U0 þ JP2 ðai aj Þ
P2 ða nÞfm ðaÞda
ð1Þ
where Unn ; Umm , and Unm are the average isotropic interaction potentials between the NPs, mesogenic molecules, and a NP and a mesogenic group, respectively. Vmm ðai aj Þ is the anisotropic interaction potential between the mesogenic molecules, which depends on the unit vectors ai and aj in the direction of the primary axis of the molecules i and j, respectively.
1 F ¼kB T1n ðln1n 1Þ þ kB T1m ðln1m 1Þ V N 1 2 1 1 U 12 U þ 1m 1n Umn 2 n nn 2 m mm Z 1 þ 12m Vmm ðai aj Þfm ðai Þfm ðaj Þdai daj 2 Z þkB T fm ðaÞlnfm ðaÞda
Z
ð4Þ
ZN ¼
Z
p
exp½b1m JSP2 ðcosgÞsingdg
ð7Þ
0
and where the nematic order parameter S satisfies the self-consistent Equation (8): S¼
1 ZN
Z
p
P2 ðcosgÞexp½b1m JSP2 ðcosgÞsingdg
ð8Þ
0
The free energy of the isotropic phase is obtained by setting S ¼ 0 [Eq. (9)]: 1 F ¼kB T1n ðln1n 1Þ þ kB T1m ðln1m 1Þ V I 1 1 12n Unn 12m Umm þ 1m 1n Umn 2 2
ð9Þ
3. Phase Separation One notes that the concentration of the NPs in the nematic phase is generally different from that in the co-existing isotropic phase, and may it be strongly temperature dependent. The co-existence between the nematic and the isotropic phases in the system under consideration is possible only if the chemical potentials of both the NPs and the mesogenic molecules are the same in the two phases. The pressure must also be the same in the two phases. However, for incompressible LCs only the equations for the chemical potentials are relevant, that is, mnI ¼ mnN and mmI ¼ mmN , where mnI and mnN are the chemical potentials of the NPs in the isotropic and in the nematic phase, respectively; and mmI and mmN are the corresponding chemical potentials of the mesogenic molecules. Using the well-known general equation for the chemical potential one obtains the following system [Eq. (10)] of two simultaneous equations: 1 @FI 1 @FN 1 @FI 1 @FN ¼ , ¼ VI @1n VN @1n VI @1m VN @1m ChemPhysChem 0000, 00, 1 – 7
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Substituting Equations (9) and (6) for the free energies of the isotropic and the nematic phase into Equation (10) one obtains the Equation (11): 1 ln mN ¼ U1 ð1mN 1mI Þ þ U12 ð1nN 1nB Þ þ lnZN 1mI
1nN ¼ U2 ð1nN 1nI Þ þ U12 ð1mN 1mI Þ 1nI
ð12Þ
Neglecting a small density change at the transition, the number densities of both the NPs and the mesogenic groups in the nematic and in the isotropic phase can be expressed in terms of the volume fraction i of the NPs in the corresponding phase i [Eq. (13)]: ð13Þ
where i ¼ N; I, 1m0 is approximately equal to the number density of the mesogenic groups in the pure LC, and 1n0 can be estimated as 1n0 :1=vn , where vn is the NP volume. Now Equations (11) and (12) can be expressed in terms of the two variables N and I [Eqs. (14, 15)]: ln
1 N ¼ w1 ðI N Þ þ lnZN 1 I
ð14Þ
ln
I ¼ w2 ðI N Þ N
ð15Þ
where [Eq. (16)]: w1 ¼ ð1m0 U1 1n0 U12 Þ, w2 ¼ ð1n0 U2 1m0 U12 Þ
ð16Þ
One notes that Equations (14) and (15) contain only two constants, w1 ; w2 , and it can readily be shown that the phase co-existence is possible only if w2 > 1. If the isotropic attraction between the NPs and the mesogenic molecules is much stronger than that between the mesogenic molecules, the inequality 1n0 U12 > 1m0 U1 is satisfied, and hence w1 < 0. The numerical solution of Equations (14), (15) together with (7) and the self-consistent Equation (8) for the nematic order parameter is significantly simplified if the volume fraction of the NPs is sufficiently small in both phases, that is, I ¼1 and N ¼1. In this case Equation (14) is simplified as [Eq. (17)]: ð1 w1 ÞðI N Þ ¼ lnZN
ð17Þ
One notes also that the partition function ZN is independent of I , and therefore I can be excluded from the system of simultaneous equations for I ¼1 and N ¼1 which results in 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
lnZN N ð1 w1 Þ
ð18Þ
Now N can be found by solving Equation (18) numerically, and I can then be evaluated in terms of N as Equation (19):
where we have introduced the non-dimensional interaction eff constants U1 ¼ Umm =ðkTB Þ, U2 ¼ Ueff nn =ðkTB Þ, U12 ¼ Unm =ðkTB Þ, and the number densities of the mesogenic molecules (1mI and 1mN ) and the NPs (1nI and 1nN ) in the isotropic and nematic phases correspondingly.
1ni ¼ 1n0 i , 1mi ¼ 1m0 ð1 i Þ
ZNw2 =ð1w1 Þ ¼ 1 þ
ð11Þ
and Equation (12) ln
a single equation for N [Eq. (18)]:
I ¼ N þ
lnZN 1 w1
ð19Þ
One notes that the volume fractions of the NPs in the nematic and the isotropic phases are not completely independent. Indeed, in the experiment one normally controls the total number of the NPs, Nn , in the volume V which yields the average volume fraction of the NPs, . It follows from the conservation of the total number of the NPs that V ¼ I VI þ N VN , where V ¼ VI þ VN is the total volume of the system. On the other hand, solutions of Equations (14) and (15) are independent of . From these equations one obtains the following expressions for the volumes VN and VI [Eq. (20)]: VI ¼ V
I , V ¼V N I N I N N
ð20Þ
One can readily see from Equation (20) that the phase co-existence is possible only if N < < I as N < I . Taking into account that N and I are independent of , one concludes that if is outside the interval ðN ; I Þ, the phase co-existence is impossible and only one phase may be stable at a given temperature. It should also be taken into account that the co-existing nematic and isotropic phases are globally stable only if the total free energy of the phase-separated system is lower than the free energy of both the isotropic and the nematic homogeneous phases. This condition can be expressed as Equation (21): sep FNI Fhom 1 VI VN ¼ F ð Þ þ FN ðN Þ FI;N ðÞ < 0 VkB T V I I VkB T V
ð21Þ
where VI and VN are given by Equation (20) and the free energy densities FI ðÞ and FN ðÞ can be expressed as Equations (22) and (23) using Equations (6), (9), and (13): FI ðÞ=VkB T ¼1n0 ln þ 1m0 ð1 Þlnð1 Þ 1 1 12n0 2 U2 12m0 ð1 Þ2 U1 þ 1m0 1n0 ð1 ÞU12 2 2 ð22Þ FN ðÞ=VkB T ¼FI ðÞ=VkB T 1 12m0 ð1 Þ2 J* S2m 1m0 ð1 ÞlnZN 2
ð23Þ
where J* ¼ J=kB T. Note that for the separated state in Equation (21) the nematic order parameter is SðN Þ in the nematic phase, which co-exists with the isotropic one, and is given by ChemPhysChem 0000, 00, 1 – 7
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CHEMPHYSCHEM ARTICLES Equation (8) with 1m ¼ 1m0 ð1 N Þ and ZN ¼ ZN ðN Þ. At the same time, the order parameter of the homogeneous nematic phase SðÞ is given by the same equation with 1m ¼ 1m0 ð1 Þ and ZN ¼ ZN ðÞ. Note that evaluation of Equation (21) can be considerably simplified if the volume of a NP is significantly larger than that of a mesogenic group, and therefore 1n =1m ¼1.
4. Phase Diagrams with Co-Existing Nematic and Isotropic Phases Let us first consider Equations (14) and (15), which enable one to determine the molar fractions of the NPs in co-existing phases as functions of temperature. A typical numerical solution of these equations for moderately strong interactions between the NPs and the mesogenic molecules is presented in Figure 1 One notes that there exists a bifurcation point which
www.chemphyschem.org system phase separates, because the separated state is globally stable only if the total free energy of the separated system is lower than that of any homogeneous phase at the same temperature and the same NP concentration [see Eq. (21)]. One notes also that if the value of is too small or too large (see the dotted line ¼ 0:01 in Figure 1), there may be no intersection at all, that is the condition N < < I is not satisfied at any temperature and, therefore, the system never phase-separates. The solutions for N and I together with Equation (21) were used to compose the temperature–concentration phase diagram presented in Figure 2 for the same values of the interac-
Figure 2. Phase diagram of the composite calculated from the mean-field theory with interaction constants w1 ¼ 5 and w2 ¼ 10.
Figure 1. NP volume fractions in co-existing isotropic (upper curve) and nematic (lower curve) phases of the composites with w1 ¼ 5 and w2 ¼ 10.
corresponds to a critical temperature Tc . At higher temperatures there is no solution, that is, the phase separation is impossible, and directly below the critical temperature the difference between the molar fractions of the NPs in the two phases is fairly small. It should be noted, however, that in the general case only part of this solution corresponds to an actual physical state of the system. As discussed in the previous section, Equations (14) and (15) do not depend on the average molar fraction of the NPs , which can be controlled experimentally. At the same time, the value of must lie between the two branches in Figure 1, as N < < N . If is different from the critical value of c at the bifurcation point, the system separates into the nematic and the isotropic phases with a finite difference between the NP molar fractions and finite volumes of the two phases. In this general case the separation occurs at some temperature Tsep which is below the bifurcation temperature Tc , and which is an intersection of the horizontal line and one of the curves representing N ðTÞ or I ðTÞ (see the intersection of N ðTÞ and the horizontal dashed line ¼ 0:04 in Figure 1). In principle, even if T < Tsep one cannot conclude that the 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim
tion constants w1 and w2 as in Figure 1. One notes that there is no phase separation at sufficiently low concentration of the NPs. In this domain, the N–I transition temperature decreases with increasing owing to the dilution effect, which has been considered in detail in ref. [18]. Above a certain critical concentration, the N–T phase transition is accompanied by the separation between the isotropic and the nematic phase, and the two phases co-exist over a significant temperature interval. In this region the transition temperature of the phase-separated state decreases more slowly than the N–I transition temperature at low concentrations. In this case, another critical value of the NP molar fraction shows up as f * , shown in Figure 2. If f > f * , the system undergoes a direct transition from the isotropic into the the phase-separated state, in which the isotropic and the nematic phases co-exist. In contrast, if f < f * , the system first undergoes a transition into the homogeneous nematic phase and then phase-separates at a somewhat lower temperature. The solution of Equations (14) and (15) and the corresponding phase diagrams for a nematic composite with very strong interaction between the NPs and the mesogenic molecules are presented in Figures 3 and 4, respectively. In this case the temperature range of the co-existence between the nematic and the isotropic phase is more narrow, and there is only a relatively small NP concentration window, in which the separation takes place at all. One notes that in this case a reentrant homogeneChemPhysChem 0000, 00, 1 – 7
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5. Discussion
Figure 3. The same as in Figure 1 with w1 ¼ 10 and w2 ¼ 30.
Figure 4. The same as in Figure 2 with w1 ¼ 10 and w2 ¼ 30.
ous nematic phase may occur within a narrow interval of the NP molar fraction. In contrast, for weaker interactions between the NPs and the mesogenic molecules (smaller absolute values of w1 ), the region of phase-separated state expands considerably (compare Figures 2 and 5). Decreasing absolute values of jw1 j result in a proportionate decrease of the critical total concentration * . Additionally, in this case the phase-separated state dominates over the low-temperature part of the phase diagram and even at small total NP concentrations the homogeneous
Figure 5. The same as in Figure 2 with w1 ¼ 1 and w2 ¼ 10.
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We have developed a mean-field molecular theory of nematic LCs doped with spherical NPs taking into account a possibility of phase separation between the nematic and the isotropic phases and considering strong isotropic interactions between the NPs and the mesogenic molecules The results appear to be rather unusual and in any case different from what one may expect from the behavior of mixtures of nematics with small non-mesogenic dopants. In particular, the temperature range of the co-existing nematic and isotropic phases may be very large. It is well known that in mixtures of LCs the N–I coexistence region is typically of the order of 1–2 degrees and can often be neglected. In contrast, the theory indicates that in nematic nano-composites the co-existence region may be as broad as the nematic phase itself. Secondly, the phase separation does not occur at sufficiently low NP concentrations because the mixing entropy is dominating the behavior of the system. Moreover, within a certain range of the NP concentration the composite may undergo a direct transition from the isotropic phase into the phase-separated state, while at other NP concentrations the system first undergoes a transition into the homogeneous nematic phase and then into the phase-separated state at lower temperature. In nano-composites with very strong interactions between the NPs themselves and between the NPs and the mesogenic molecules, the N–I phase separation may only occur within a finite NP concentration interval, and a reentrant homogeneous nematic phase may be stable for some concentrations. These unexpected results are mainly related to the fact that the properties of the NPs differ very much from those of typical mesogenic molecules. In particular, the isotropic interactions between the NPs and the mesogenic groups (and between the NPs themselves) is expected to be significantly stronger than those between the mesogenic molecules. This is mainly due to the large effective volume of a typical spherical NP, which also includes organic chains attached to the surface of the metal or semiconductor core. As discussed in detail in this paper, the total free energy may be minimized if the system separates into the isotropic phase with an increased NP concentration and the nematic phase with a lower NP concentration. This kind of phase separation has been observed in few anisotropic soft-matter systems.[2, 6] However, to the best of our knowledge, it has not been studied in detail experimentally so far. Very recently, however, a two-step decrease of both the N–I transition temperature and the transition heat in few polymer and low molecular weight LCs doped with quantum dots[22] has been interpreted assuming that the system separated in to the isotropic and the nematic phase with different concentrations of quantum dots. A different type of phase separation has recently been observed in refs. [27, 28]. It has been shown experimentally that gold NPs with mesogenic coatings form reversible networks composed of nematic droplets accompanied by disclination ChemPhysChem 0000, 00, 1 – 7
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CHEMPHYSCHEM ARTICLES lines and loops as a result of a specific phase separation along with the enrichment of NPs at the N–I liquid interfaces. In addition, the properties of nematic LCs doped with metal, dielectric, or semiconductor NPs differ significantly from those of nematics with relatively large colloidal particles. For example, it has been shown theoretically in ref. [29] that in nematics doped with colloidal particles phase separation at the N–I transition occurs at all particle concentrations, that is, there has been no threshold concentration found. This has been explained by the large size of the colloidal particles, which produce director distortions in the surrounding LC medium. Such director distortions have been modeled in ref. [29] using a random anisotropic field which results in a different mathematical form of the free energy of the mixture. As a result the tendency for the separation at low concentrations of colloidal particles is strongly enhanced. A very broad N–I co-existence region in LCs doped with colloidal particles has also been described in refs. [24, 25] using simple mean-field theory of mixtures. This theory, however, is based on a different (and rather oversimplified) model interaction potential that is more suitable for large colloidal particles. In our paper we have not taken into consideration possible formation of chains and other types of NP aggregation, which, in principle, may strongly modify the conditions for N–I phase separation. There exists some experimental evidence that quantum dots may form long chains in nematic LCs.[26] Large values of the dielectric anisotropy in the nematic phase doped with ferroelectric particles have also indicated that strongly polar particles may form chains.[16] Chain ordering of NPs thus definitely deserves a separate investigation.
Acknowledgements This work was supported by the Ministry of Education and Science of Russian Federation (agreement 8509) and the Russian Foundation for Basic Research (project No. 13-03-00579 a). Keywords: liquid crystals · molecular theory · nanoparticles · nematic phase · phase transition [1] H. Qi, B. Kinkead, T. Hegmann, Proc. SPIE Int. Soc. Opt. Eng. 2008, 6911, 691106.
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Received: November 11, 2013 Published online on && &&, 2014
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ARTICLES Transit: A molecular-statistical theory is developed which enables one to describe the nematic-isotropic phase transition in liquid crystals doped with spherical nanoparticles taking into account the effects of phase separation.
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M. A. Osipov,* M. V. Gorkunov && – && Molecular Theory of Phase Separation in Nematic Liquid Crystals Doped with Spherical Nanoparticles
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