Physica A Precursors of long-range order and local disorder in colloids

M. Mayorga et al. / Physica A 388 (2009) 1973–1977 1975Fig. 1. Pair correlation function g(r) for our model potential plotted against reduced distance r ∗ . At the packing fraction η = 0.494 and η = 0.545 wecan observe the appearance of the precursor and the presence of a structural long range order, respectively. The shoulder is practically flat, thus indicatingthat particles have a higher degree of packing. The inset enhances the split which resembles the structural precursor to freezing observed in hard-spheresystems [16].Fig. 2. This image shows the translational long range order for the colloidal particles at η = 0.545, for the values of the parameters that were used toobserve the freezing precursor.energy barrier corresponding to the clustering process, and consequently to a decrease of the surface tension in accordancewith the classical theory of nucleation [13].The precursor of freezing is manifested through the shoulder that appears on the left hand side of the second peak inFig. 1, in the range of reduced distances (1.85, 1.95) which is included in the corresponding interval found in Figure 6 ofRef. [16], obtained in the case of hard spheres. The shoulder, identical to the one reported for the hard sphere system at thesame packing fraction, namely η = 0.494, with α = 0.945 and β = 14.15, is the signature of the formation of crystallineclusters and thus constitutes a crystalline-structure precursor, in addition. For each time-step, we calculate the position ofthe particles from which we obtain the snapshot of the system showing the presence of spatial and orientational order, aswe can see in Figs. 2 and 3. Once the precursor arises, cluster formation becomes favored by fixing the value of β. Increasingthe packing fraction of the system at the value η = 0.545, we obtain a tightly-packed hexagonal-closed array. This featurebecomes manifest in the dashed line in Fig. 1. On the other hand, for the above values of the potential parameters, theeffective width of the attractive well is less than one-third of the particle’s diameter. We observe, in addition, that thefreezing precursor is still observed when the parameter β increases, i.e. decreasing the ratio effective width/diameter. Thislast situation is of interest to colloidal systems with a very short-range attraction for which it is well known that there existonly two stable phases: fluid and crystal [19], or for colloids with an even narrower attractive well, with two solid phasesthat differ only in density [20].Following a protocol similar to the one previously outlined, we observe a transition from an ordered structure forβ = 11.910 to the formation of amorphous clusters for β = 11.905 maintaining the values α = 0.945 and the packing

1976 M. Mayorga et al. / Physica A 388 (2009) 1973–1977Fig. 3. This image shows a rotation of Fig. 2 from which one can better observe the presence of orientational long range order. The packing fraction is alsoη = 0.545 which corresponds to the same conditions given in Fig. 2 but the figure has been slightly rotated to capture the corresponding view.Fig. 4. Pair correlation function for our model potential plotted versus reduced distance r ∗ , for two different β parameters. The continuous line is forβ = 11.910 and β = 11.905 for the dash line, for both cases α = 0.945 and η = 0.494. In this case, the shoulder is no longer flat. The particles becomearrested, giving rise to disordered configurations which have been found experimentally. The inset enhances the shoulder of the continuous plot whichshows qualitative agreement with the glassy precursor obtained experimentally in Refs. [17,21].fraction η = 0.494. The corresponding correlation functions are shown in Fig. 4. This function is similar to the one observedin experiments for PMMA particles immersed in a mixture of decalin and cycloheptylbromide, that were reported in theFigure 4(b) of Ref. [17]. The lower value of β corresponds to an increase of the diffusion of particles in comparison withthe ordered case. As a consequence, the shoulder in g appears at the right-hand side of the second maximum, leadingto the appearance of cages, one of the features of colloids close to glass transition [17,21], which can be considered asa sign of an amorphous-structure precursor. The above observation establishes that there is a crystallization window(i.e. β > 11.910) for which the crystallization can be induced, increasing the volume fraction. So, decreasing β belowthat value, the crystallization is frustrated, resulting in a disordered structure.4. ConclusionsWe have shown the existence of structural precursors of long-range order and local disorder in colloids, which can beidentified as shoulders around the second peak in the pair correlation function. The presence of a shoulder on the left (right)side is the signature of long-range order (local disorder) in colloidal suspensions. Its position depends on the value of asingle parameter that modulates the effective width of the attractive well in the interaction potential. These interestingfindings may contribute to a deeper understanding of the models that can be proposed to study phase transitions in colloidalsuspensions [22].

M. Mayorga et al. / Physica A 388 (2009) 1973–1977 1977AcknowledgmentsWe would like to thank S. Torquato, D. Frenkel and A. van Blaaderen for interesting discussions. We want to acknowledgefinancial support of PROMEP and grants UAEM 2263/2006, CONACyT P/49607 and by DGICYT of the Spanish Governmentunder Grant No. FIS2005-01299.References[1] K.H. Lin, et al., Phys. Rev. Lett. 85 (2000) 1770.[2] P.G. Vekilov, Cryst. Growth Des. 4 (2004) 671.[3] J.F. Lutsko, G. Nicolis, Phys. Rev. Lett. 96 (2006) 046102.[4] D. Frenkel, Nature Mater. 5 (2006) 85.[5] W.K. Kegel, A. van Blaaderen, Science 287 (2000) 290.[6] A. Yethiraj, J.H.J. Thijssen, A. Wouterse, A. Van Blaaderen, Adv. Mater. 16 (2004) 596.[7] T.F. Krauss, Nature Mater. 2 (2003) 777.[8] J. Gao, Z. Hu, Langmuir 18 (2002) 1360.[9] J. Wu, G. Huang, Z. Hu, Macromolecules 36 (2003) 440.[10] L.F. Filobelo, O. Galkin, P.G. Vekilov, J. Chem. Phys. 123 (2005) 014904;W. Pan, A.B. Kolomeisky, P.G. Vekilov, J. Chem. Phys. 122 (2005) 174905.[11] A. Ben-Naim, Hydrophobic Interactions, Springer, 1980.[12] J.N. Israelavichli, Intermolecular and Surface Forces, Academic Press, San Diego, 1995.[13] D. Kashchiev, Nucleation, Basic Theory and Applications, Butterworth-Heinemann, Oxford, 2000.[14] M. Mayorga, L. Romero-Salazar, J.M. Rubi, Physica A 307 (2002) 297.[15] D. Osorio-González, M. Mayorga, J. Orozco, L. Romero-Salazar, J. Chem. Phys. 118 (2003) 6989.[16] T.M. Truskett, S. Torquato, S. Sastry, P.G. Debenedetti, F.H. Stillinger, Phys. Rev. E 58 (1998) 3083.[17] E.R. Weeks, D.A. Weitz, Phys. Rev. Lett. 89 (2002) 095704.[18] R. Verberg, I.M. de Schepper, E.G.D. Cohen, Phys. Rev. E 61 (2000) 2967;R. Verberg, I.M. de Schepper, E.G.D. Cohen, Phys. Rev. E 55 (1997) 3143.[19] H.N.W. Lekkerkerker, W.C.K. Poon, P.N. Pusey, A. Stroobants, P.B. Warren, Europhys. Lett. 20 (1992) 559.[20] P. Bolhuis, D. Frenkel, Phys. Rev. Lett. 72 (1994) 2211.[21] E.R. Weeks, J.C. Crocker, D.A. Weitz, J. Phys. Cond. Matt. 19 (2007) 205131.[22] J.R. Savage, D.W. Blair, A.J. Levine, R.A. Guyer, A.D. Dinsmore, Science 314 (2007) 795.