Exact calculations of the paranematic interaction energy

the order of magnitude and the sign of the interaction is correct in the regime where the paranematic attraction com-petes with the electrostatic repu...

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Exact calculations of the paranematic interaction energy for colloidal dispersions in the isotropic phase of a nematogenic material J.-B. Fournier* Laboratoire de Physico-Chimie The´orique, ESPCI, 10 Rue Vauquelin, F-75231 Paris Cedex 05, France

P. Galatola† LBHP, Universite´ Paris 7–Denis Diderot, Case 7056, 2 Place Jussieu, F-75251 Paris Cedex 05, France 共Received 9 October 2001; published 5 March 2002兲 In a recent paper 关Phys. Rev. E 61, 2831 共2000兲兴, Borsˇtnik, Stark, and Zˇumer have studied the stability of a colloidal dispersion of micron-sized spherical particles in the isotropic phase of a nematogenic material. Close to the nematic transition, the attraction due to a surface-induced paranematic order can yield flocculation. Their calculation of the nematic-mediated interaction was based on an ansatz for the order-parameter profile. We compare it with an exact numerical calculation, showing that their results are qualitatively correct. Besides, we point out that in the considered regime, the exact interaction is extremely well approximated by a simple analytical formula which is asymptotically exact. DOI: 10.1103/PhysRevE.65.032702

PACS number共s兲: 61.30.Cz, 82.70.Dd

In recent years, a great deal of interest has been devoted to understanding the interactions and phase behavior of colloidal particles dispersed in a nematic phase 关1–5兴 or in the isotropic phase of a nematogenic compound 关6 –9兴. In the nematic phase, colloids experience a specific elastic interaction because they induce competing distortions of the nematic director field. New physics arise due to the long-range character of this interaction and the induction of topological defects 关1兴. In the isotropic phase, the surface of colloidal particles can induce a local paranematic order 关10,11兴, giving rise to a short-range elastic interaction 关6,7兴. Two effects compete: an attraction due to the favorable overlapping of the paranematic halos and a repulsion due to the distortion of the director field. For small particles, of size comparable to the nematic-isotropic coherence length ␰ , it has been predicted that repulsion may dominate and stabilize the colloidal dispersion 关6,9兴. 共Note that latex particles as small as 50 nm have been successfully dispersed in lyotropic nematics 关12兴.兲 On the other hand, Borsˇtnik, Stark, and Zˇumer have predicted that for micron-sized particles attraction dominates 关7兴 and should allow us to trigger flocculation close to the nematic transition 关8兴. The results of Borsˇtnik, Stark, and Zˇumer 关8兴 are based on a composite ansatz for the nematic director field n and for the scalar order-parameter Q, within an uniaxial hypothesis. It turns out that our exact calculations 关9兴, based on a multipolar expansion for the full tensorial order-parameter Q i j , rest on the same theoretical model, and can be performed also for micron-sized particles. In this Brief Report, we check the correctness of the paranematic interaction energy used in Ref. 关8兴, by comparing it with the exact one, numerically calculated according to the method of Ref. 关9兴. For the typical values considered in Ref. 关8兴, we find that the exact interaction is attractive in*Electronic address: [email protected]

Electronic address: [email protected]


stead of repulsive at distances ⲏ5 ␰ , and that it is about two times weaker at distances of the order of ␰ 共Fig. 1兲. However, the order of magnitude and the sign of the interaction is correct in the regime where the paranematic attraction competes with the electrostatic repulsion. In particular, at particles’ separations of the order of ␰ , the paranematic attraction remains much larger than the van der Waals attraction: this implies that the conclusions of Ref. 关8兴 regarding critical flocculation phenomena remain qualitatively correct. Finally, we show that in the whole regime of interest the exact interaction calculated numerically is extremely well approximated by a simple asymptotic formula 关see Eq. 共8兲兴. The Landau–de Gennes 关13兴 paranematic free-energy density considered in Refs. 关7,8兴 is 3 9 1 f ⫽ a⌬TQ 2 ⫹ L 1 共 “Q 兲 2 ⫹ L 1 Q 2 兩 “n兩 2 2 2 2


for the bulk, and f s ⫽G Q 共 Q⫺Q s 兲 2 ⫹3G n QQ s sin2 ␪


for the surface. Here, Q is the scalar order parameter and n the nematic director. The parameter Q s is the order parameter favored by the particles’ surface, and ␪ is the angle between the direction of n at the surface and the normal to the surface ␯, which is assumed to correspond to the easy axis. The coefficients a and L 1 are material parameters, ⌬T ⫽T⫺T 쐓 is the difference between the actual temperature T and the limit of stability T 쐓 of the isotropic phase, and G Q and G n are introduced to describe the strength of the surface anchoring. Incidentally, we note that the surface free-energy density 共2兲 is inconsistent within a Landau-de Gennes framework, unless G Q ⫽G n . Indeed, at quadratic order, the most general expansion in terms of the tensorial order parameter Q i j and of the normal ␯ to the surface can be written as

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©2002 The American Physical Society



with Q s ⫽⫺g 1 /3g 21 . Therefore, Eq. 共2兲 is compatible with this expression only if G Q ⫽G n . With the required condition G Q ⫽G n ⬅G, the free energy considered in Ref. 关8兴 is identical to the one we used in Ref. 关9兴, with the correspondence 关14兴 L † ⫽2L 1 ,

3 S † ⫽ Q, 2

2 a † ⫽ a⌬T, 3

FIG. 1. Paranematic interaction energy between two spherical particles of radius R⫽0.25 ␮ m as a function of their distance to contact d. The parameters are a⫽1.8⫻105 J m⫺3 K⫺1 , T * ⫽313.5 K, ⌬T⫽1.3 K, L 1 ⫽9⫻10⫺12 J m⫺1 , Q s ⫽0.3, G Q ⫽G n ⫽10⫺3 J m⫺2 . The corresponding nematic coherence length is ␰ ⫽10.7 nm. The dashed-dotted line is extracted from Fig. 7 of Ref. 关7兴. The full line is the exact result, numerically calculated according to Ref. 关9兴. The dotted line, practically coincident with the full line, corresponds to the asymptotic formula 共8兲.

f s ⫽g 1 Q i j ␯ i ␯ j ⫹g 21Q i j Q i j ⫹g 22Q i j Q jk ␯ i ␯ k ⫹g 23Q i j Q kl ␯ i ␯ j ␯ k ␯ l ,


F⫽⫺48␲ L 1 ␰ 共4兲

4 W † ⫽ G, 3

共7a兲 共7b兲

where we have daggered the quantities appearing in Ref. 关9兴. With the above relationships, we have numerically recalculated the exact interaction energy between two spherical particles of radius R as a function of their distance to contact d—using the same parameters as in Ref. 关7兴. The comparison with the results of the ansatz of Borsˇtnik, Stark, and Zˇumer 关7兴 is shown in Fig. 1. We find a qualitative agreement, as previously discussed. Note that the exact nematic director profile displays a ring defect 关9兴, which is absent in the ansatz of Ref. 关7兴. For the case of micron-sized particles considered in Refs. 关7,8兴, this defect lies however in a region where the paranematic is almost completely melted. Finally, we have compared our numerical calculation with the asymptotic formula obtained by us in Ref. 关9兴. With the correspondence 共7兲, the latter reads

where summation over repeated indices is implied. For a uniaxial tensorial order parameter 3 1 Q i j⫽ Q n in j⫺ ␦ i j , 2 3

3 S †0 ⫽ Q s , 2

冉 冊 Qs A



e ⫺d , ¯ ⫹d ¯ 2R


where ¯d ⫽d/ ␰ , ¯R ⫽R/ ␰ , and A is a constant given by

this yields 1 3 f s ⫽g 1 Q⫹ 关 3g 21⫹2 共 g 22⫹g 23兲兴 Q 2 ⫺ 关 2g 1 Q 2 4


27l¯ 6⫹27l¯ 6⫹12l¯ 2⫹3 l¯ ⫹ ⫹ ⫹ , ¯R 4 ¯R 3 ¯R 2 ¯R



where l¯ ⫽L 1 /G ␰ is the reduced extrapolation length of the anchoring. The nematic coherence length is ␰ ⫽ 冑3L 1 /a⌬T. As shown in Fig. 1, the agreement between the numerical and the analytical calculations is excellent in the range of separations relevant to the colloidal flocculation discussed in Ref. 关8兴. Owing to its simplicity and validity, formula 共8兲 thus offers a straightforward means to systematically investigate the stability of such paranematic-wetted colloids.

关1兴 P. Poulin, H. Stark, T.C. Lubensky, and D.A. Weitz, Science 275, 1770 共1997兲. 关2兴 P. Poulin, V. Cabuil, and D.A. Weitz, Phys. Rev. Lett. 79, 4862 共1997兲. 关3兴 H. Stark, J. Stelzer, and R. Bernhard, Eur. Phys. J. B 10, 515 共1999兲. 关4兴 R. Yamamoto, Phys. Rev. Lett. 87, 075502 共2001兲. 关5兴 D. Andrienko, G. Germano, and M.P. Allen, Phys. Rev. E 63,

041701 共2001兲. 关6兴 P. Galatola and J.-B. Fournier, Mol. Cryst. Liq. Cryst. 330, 535 共1999兲. 关7兴 A. Borsˇtnik, H. Stark, and S. Zˇumer, Phys. Rev. E 60, 4210 共1999兲. 关8兴 A. Borsˇtnik, H. Stark, and S. Zˇumer, Phys. Rev. E 61, 2831 共2000兲. 关9兴 P. Galatola and J.-B. Fournier, Phys. Rev. Lett. 86, 3915

9 ⫹ 共 g 22⫹4g 23兲 Q 2 兴 sin2 ␪ ⫹ g 23Q 2 sin4 ␪ . 4


Matching Eqs. 共5兲 and 共2兲 requires setting g 23⫽0 and consequently g 22⫽0. Equation 共5兲 then becomes f s ⫽⫺

g 12 3 ⫹ g 21关共 Q⫺Q s 兲 2 ⫹3QQ s sin2 ␪ 兴 , 6g 21 2




共2001兲. 关10兴 P. Sheng, Phys. Rev. Lett. 37, 1059 共1976兲. 关11兴 K. Miyano, Phys. Rev. Lett. 43, 51 共1979兲. 关12兴 O. Mondain-Monval, J.C. Dedieu, T. Gulik-Krzywicki, and P. Poulin, Eur. Phys. J. B 12, 167 共1999兲.

关13兴 P.-G. de Gennes and J. Prost, The Physics of Liquid Crystals 共Clarendon, Oxford, 1993兲. 关14兴 Actually, in our free energy we keep the full biaxial orderparameter Q i j instead of assuming the uniaxial order 共4兲.