Article pubs.acs.org/Langmuir
Short-Time Dynamic Signature of the Liquid−Crystal−Glass Transition in a Suspension of Charged Spherical Colloids P. Holmqvist* ICS-3, Forschungszentrum Jülich, Postfach 1913, 52425 Jülich, Germany ABSTRACT: In this paper, the dynamic transition of the liquid− crystal−glass transition is investigated by dynamic light scattering, DLS. From the intensity autocorrelation function, g2(q, t), the short-time dynamic function, D(q), has been determined at different concentrations in both the crystal and glass regions. From D(q), the short-time self-diffusion, ds, was determined. ds speeds up in the crystal state but has very similar characteristics in the liquid and the glass region. The general model in which the colloidal crystallization transition in a spherical colloidal system is driven by an increase in local entropy is also verified by relating ds to the local excess entropy. Experimentally determined structure factors, S(q), are also discussed, and we show the similarity between the glass and the liquid. This investigation shows that the liquid−crystal transition can be identified in addition to the appearance of Bragg peaks with a short-time dynamic transition while no sharp transition in the short-time dynamics or S(q) can be found between the glass and the liquid.
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INTRODUCTION Suspensions of spherical colloids have been used with great success to model the static and dynamic properties of atomic fluids.1 An important and interesting phenomenon in colloidal as well as atomic systems is the liquid to solid transition, including the transition to a glass. For atomic liquids, this transition depends on the temperature and pressure while for colloidal systems it can additionally be moderated by the colloidal interactions and concentration. The interactions, both the attractive and the repulsive, can be easily tuned in colloidal systems. Attractive interactions, apart from the van der Waals interaction, is most commonly introduced by depletion2−5 while repulsive interactions can be added to the colloids by steric repulsion via polymer brushes on the surface or by electrostatic repulsion.6 For reasonably low polydisperse charged spherical colloids in a low-salinity solvent, the liquid−solid transition is a liquid to crystal phase transition. In order to get such a system in a glassy state, one has to increase the concentration beyond the crystallization regime. The liquid−crystal transition can be identified optically by the formation of iridescent crystallites while Bragg peaks will be found in static scattering experiment. The crystal−glass phase transition can in principle be determined in the same way, but since the crystallites becomes smaller and smaller, the transition is much harder to determine. The dynamic parameters, the relaxation processes, in colloidal suspensions are also affected by the phase transitions and are therefore useful parameters for the determination of the phase boundaries. In the liquid state of charged spherical colloids, the short-time diffusive behavior is well understood, and very good agreement has been established between theory, simulations, and experiments.7−12 The prediction for the long-time diffusion in the liquid regime of hard-sphere colloidal suspensions has also been reported.13,14 © 2014 American Chemical Society
At the freezing concentration corresponding to the liquid to solid transition, the long-time diffusion coefficient, DL, has slowed 10 times more than its short-time quantity, which was suggested by Löwen et al. to be a universal behavior.15 A substantial amount of investigation addresses the long-time diffusion and the nonergodicity parameter, g1(q, ∞), as the glass transition is approached.16−19 In the glass state, DL is in principle zero and the system is considered to be nonergodic. For colloidal crystals, the measured correlation functions are still ergodic and a long time tail whose origin is not well established can be found. A less addressed issue, in general, is how the short-time dynamics relates to phase transitions, specifically for charged-stabilized colloidal suspensions. The investigation of the short-time equilibrium relaxation process and the effect the phase transitions has on this process will also shed light on the crystallization and melting behavior of longrange repulsive colloidal systems. While most experimental investigations of crystal and glass phases in long-ranged repulsive systems address static and kinetic properties such as melting and crystal formation,20,21 the focus of this paper is on the short-time equilibrium dynamics. An interesting question is how the short-time motion is reflecting the local entropy in the system since there is a loss of entropy when a colloidal suspension freezes into a crystal and the long-range positional order is increased. It is possible to access the entropic gain at a small length scale from short-time dynamic measurements using an empirical relation between the self-diffusion and the excess entropy.22−24 Received: March 21, 2014 Revised: May 28, 2014 Published: May 28, 2014 6678
dx.doi.org/10.1021/la5010853 | Langmuir 2014, 30, 6678−6683
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In this paper, the short-time diffusive response to the liquid− crystal and crystal−glass transition has been investigated as well as its dependence on concentration in these phases and the wave vector, q, dependence. The short-time self-diffusion, ds, is determined from the measurements since it is the only dynamic property from dynamic light scattering, DLS, which can be continuously investigated in a consistent and well-established way over phase boundaries and through the different phases. The main reason is that in a crystal there is no established procedure for the determination of DL from the intensity autocorrelation function, g2(q, t), and in a glass, DL slows down beyond the measuring time (nonergodicity). We will present experimentally determined structure factors, S(q), which will be briefly discussed together with the corresponding dynamic functions, D(q), in both the crystal and glass regimes of charged colloidal suspensions. From this it will be shown how the shorttime self-diffusion rate, ds, changes from the liquid to the crystal to the glass phase. Its consequences on the local excess entropy will also be discussed.
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Figure 1. g1(q, t) at a wave vector of 0.028 nm−1 for ϕ = 0.186 (filled squares), ϕ = 0.217 (filled circles), ϕ = 0.302 (open squares), ϕ = 0.350 (open circles), and ϕ = 0.420 (open triangles). The inset is the same data as represented by g2(q, t) − 1.
EXPERIMENT baseline will then be on the order of 10−1, which is clearly visible (Figure 1). Since the small baseline in g2(q, t) does not affect the extracted relaxation times, the measurements are considered to be ergodic for all samples and q vectors in the crystalline region. This is confirmed by testing different techniques’ analytical approaches for nonergodic27,28 and ergodic measurements, which all gave the same g1(q, t). The short-time diffusive function D(q) = 1/τ(q)q2 was determined by fitting g1(q, t) in the crystal regime in the same manner as was done in the liquid regime.26 The correlation functions are considered to be fully relaxed and consist of one relaxation process. The short-time relaxation times are determined by the initial slope, as was done, for example, in neutral-sphere systems.29,30 Other analytical methods used to analyze the data have been tried in order to establish the best approach. For example, a stretched exponential fitting procedure gives a similar result but does not take into account the long-time relaxation part of the correlation function, and CONTIN or multiexponential fits are used by assuming two or more distinctly different processes which are not the case in this system. On the other hand, the samples in the glass regime (open symbols in Figure 1) show a clear baseline represented by both g1(q, t) and g2(q, t) − 1 (inset). In the glass region, the baseline in g2(q, t) − 1 is above 0.05 and in g1(q, t) it is above 0.2, and two single measurements at different positions in the sample do not give the same g2(q, t) − 1. This shows that the samples are nonergodic. Thus, all measurements in this region were made using the brute force measurements by averaging over 300 correlation functions (no significant difference could be seen when averaging over 600 correlation functions for several measurements). The baseline in g1(q, t) is commonly known as the nonergodicity factor, g1(q, ∞). In the glass regime, g1(q, ∞) increases with increasing concentration. To investigate the dynamic response in the glass, the shorttime diffusive function D(q) was determined by fitting the initial slope of g1(q, t) with an added baseline defined as g1(q, ∞) = 1 − A, where A is the amplitude of the process. In the glass samples, the intermediate time regime in the correlation functions, which is the regime between the first decay and the plateau, cannot be described with a stretched exponential function. It can, however, be analyzed with the power law decay
The investigation was performed with dynamic, DLS, and static, SLS, light scattering on charge-stabilized trimethoxysilypropyl methacrylate (TPM)-coated silica spheres25 in an index-matching 80/20 toluene/ ethanol mixture at T = 20 °C and a Bjerrum length of LB = e2/ (4πε0εrkBT) = 8.64 nm, where e is the elementary charge, ε0 is the vacuum permittivity, εr is the relative dielectric constant, kB is the Boltzmann constant, and T is the temperature. Due to the absence of uncontrolled CO2 adsorption, this mixture has a residual salinity of ns = 0.7 μM.26 The radius determined by small-angle X-ray scattering, SAXS, is a = 136 nm, with a size polydispersity of 0.06. The measurements of the static structure factor, S(q), and the dynamic function, D(q), were made in a broad wave vector, q, range at concentrations above the freezing transition. The light-scattering measurements were performed on an ALV-Laservertriebsgesellschaft (Langen, Germany), and the intensity autocorrelation function, g2(q, t), was recorded with an ALV-5000 multitau digital correlator. The electric field autocorrelation function, g1(q, t), was determined in the ergodic case from g2(q, t) using the Siegert relation g2(q , t ) − 1 = (g1(q , t ))2
(1)
and in the nonergodic cases by the so-called brute force method where numerous g2(q, t) were measured at different positions in the samples and then averaged according to g1(q , t ) = N
∑i [g2(q , t )i − 1]Ii ,tot 2 (∑i Ii ,tot)2
(2)
where N is the number of measurements and Ii,tot the total time average intensity of the ith measurement. We checked that there is no noticeable multiple scattering in any sample.
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RESULTS Correlation Function. Figure 1 shows g1(q, t) at a wave vector of 0.028 nm−1 for two samples in the crystal region (solid symbols). As is evident from the figure, the field autocorrelation functions g1(q, t) in the crystal-phase region have a small baseline, qmax where S(q) = 1. At the highest concentrations where this point was outside the measuring range, the last largest measured q value was used. This will give a value of ds that is slightly too low, but the error is still reasonably small. Also, the cage diffusion, D(qmax), was determined as the value of D(q) where S(q) peaks at S(qmax). The normalized diffusion coefficients ds/d0 and D(qmax)/d0 are displayed in Figure 4 as a function of volume fraction. In the liquid region, the samples show no Bragg reflection (Figure 4), and both ds and D(qmax) following the expected trends reported in refs 9 and 26, where they monotonically decrease with increasing concentration. As described in ref 9, the concentration dependence of ds is distinctly different from the hard-sphere prediction35 but is in very good agreement with the prediction for charged spherical colloids, ds/d0 = 1 − 2.5ϕ4/3.12 Interestingly, as the system enters the crystal regime ϕ ≈ 0.16 (represented by a horizontal dotted line in Figure 4), the short-time dynamics speeds up by almost a factor of 2. This increase in ds over the liquid to crystal phase transition represents an increase in the local excess entropy, ΔSexc, which can be empirically expressed as ln(ds) ≈ ΔSexc.22−24 It is now apparent that due to the ordering in the crystal the colloidal particles gain an increase in local freedom represented by an increase in ΔSexc. The finding confers the general description of the entropy-driven crystallization as described for example by Pusey36 and Ackerson.37 Since there are no, or very small, enthalpy gains in the colloidal crystallization, different from many atomic crystallization processes, the only remaining factor is the entropy. At first glance this seems to be contradictory since increasing order means a decrease in entropy. But due to the global increase in order more space is available for every individual colloid and the local disorder is increased. Therefore, the local entropy is increased, and is ds. The present data verify that there is a local increase in the entropy due to the crystallization by the increase in ds. The use of the ln(ds) ≈ ΔSexc relation is valid if ds can be related to the pair-correlation function, g(r). For the present case, the assumption holds since ds is related to g(r) through the hydrodynamic function. In addition, ds and D(qmax) become equal in the crystal state since no slowing down at qmax occurs. As the sphere concentration is increased further in the crystal region, the crystallites become smaller and smaller (Figure 4) while the dynamics slows slightly. Then as the colloidal concentration is increased over the glass transition, the dynamic slows down substantially, indicating a loss of local freedom which together with the lack of Bragg reflections and nonergodic correlation functions identifies the state as a frozen liquid. Interestingly enough, the level at which the short-time diffusion and the short-time cage diffusion slow to in the glassy state seems to be the extension of 6681
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(8) Heinen, M.; Banchio, A. J.; Nägele, G. Short-time rheology and diffusion in suspensions of Yukawa-type colloidal particles. J. Chem. Phys. 2011, 135, 154504. (9) Heinen, M.; Holmqvist, P.; Banchio, A. J.; Nägele, G. Short-time diffusion of charge-stabilized colloidal particles: generic features. J. Appl. Crystallogr. 2010, 43, 970−980. (10) Heinen, M.; Holmqvist, P.; Banchio, A. J.; Nägele, G. Pair structure of the hard-sphere Yukawa fluid: An improved analytic method versus simulations, Rogers-Young scheme, and experiment. J. Chem. Phys. 2011, 134, 129901. (11) Heinen, M.; Holmqvist, P.; Banchio, A. J.; Nägele, G. Pair structure of the hard-sphere Yukawa fluid: An improved analytic method versus simulations, Rogers-Young scheme, and experiment. J. Chem. Phys. 2011, 134, 044532. (12) Banchio, A. J.; Nägele, G. Short-time transport properties in dense suspensions: From neutral to charge-stabilized colloidal spheres. J. Chem. Phys. 2008, 128, 104903. (13) Banchio, A. J.; Bergenholtz, J.; Nägele, G. Rheology and dynamics of colloidal suspensions. Phys. Rev. Lett. 1999, 82, 1792− 1795. (14) Banchio, A. J.; Nägele, G.; Bergenholtz, J. Collective diffusion, self-diffusion and freezing criteria of colloidal suspensions. J. Chem. Phys. 2000, 113, 3381−3396. (15) Löwen, H.; Palberg, T.; Simon, R. Dynamic Criterion for Freezing of Colloidal Liquids. Phys. Rev. Lett. 1993, 70, 1557−1560. (16) Härtl, W.; Versmold, H.; Zhangheider, X. The Glass-Transition of Charged Polymer Colloids. J. Chem. Phys. 1995, 102, 6613−6618. (17) Joosten, J. G. H.; Mccarthy, J. L.; Pusey, P. N. Dynamic and Static Light-Scattering by Aqueous Polyacrylamide Gels. Macromolecules 1991, 24, 6690−6699. (18) van Megen, W.; Underwood, S. M. Dynamic-Light-Scattering Study of Glasses of Hard Colloidal Spheres. Phys. Rev. E 1993, 47, 248−261. (19) van Megen, W.; Underwood, S. M. Glass-Transition in Colloidal Hard-Spheres - Mode-Coupling Theory Analysis. Phys. Rev. Lett. 1993, 70, 2766−2769. (20) Alsayed, A. M.; Islam, M. F.; Zhang, J.; Collings, P. J.; Yodh, A. G. Premelting at defects within bulk colloidal crystals. Science 2005, 309, 1207−1210. (21) Royall, C. P.; Leunissen, M. E.; van Blaaderen, A. A new colloidal model system to study long-range interactions quantitatively in real space. J. Phys.: Condens. Matter 2003, 15, S3581−S3596. (22) Dzugutov, M. A universal scaling law for atomic diffusion in condensed matter. Nature 1996, 381, 137−139. (23) Kaur, C.; Harbola, U.; Das, S. P. Nature of the entropy versus self-diffusivity plot for simple liquids. J. Chem. Phys. 2005, 123, 034501. (24) Rosenfeld, Y. A quasi-universal scaling law for atomic transport in simple fluids. J. Phys.: Condens. Matter 1999, 11, 5415−5427. (25) Philipse, A. P.; Vrij, A. Determination of Static and Dynamic Interactions between Monodisperse, Charged Silica Spheres in an Optically Matching, Organic-Solvent. J. Chem. Phys. 1988, 88, 6459− 6470. (26) Holmqvist, P.; Nägele, G. Long-Time Dynamics of Concentrated Charge-Stabilized Colloids. Phys. Rev. Lett. 2010, 104, 058301. (27) Pusey, P. N.; van Megen, W. Dynamic Light-Scattering by NonErgodic Media. Physica A 1989, 157, 705−741. (28) Xue, J. Z.; Pine, D. J.; Milner, S. T.; Wu, X. L.; Chaikin, P. M. Nonergodicity and Light-Scattering from Polymer Gels. Phys. Rev. A 1992, 46, 6550−6563. (29) Segrè, P. N.; Meeker, S. P.; Pusey, P. N.; Poon, W. C. K. Viscosity and Structural Relaxation in Suspensions of Hard-Sphere Colloids. Phys. Rev. Lett. 1995, 75, 958−961. (30) Segre, P. N.; Pusey, P. N. Scaling of the dynamic scattering function of concentrated colloidal suspensions. Phys. Rev. Lett. 1996, 77, 771−774. (31) Gotze, W.; Sjogren, L. Beta-Relaxation at the Glass-Transition of Hard-Spherical Colloids. Phys. Rev. A 1991, 43, 5442−5448. (32) Gotze, W.; Sjogren, L. Relaxation Processes in Supercooled Liquids. Rep. Prog. Phys. 1992, 55, 241−376.
the trend in the liquid state. It indicates that if the crystal regime should not have been in between the liquid and glass regimes then no short-time dynamic transition would have been observed.
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CONCLUSIONS In this paper, the dynamic transition of the liquid−crystal−glass phase transition of a suspension of charged spherical colloids was investigated by measuring the short-time dynamic function for different sphere concentrations in both the crystal and glass regions. It has been shown that the short-time self-diffusion speeds up in the crystal state. By relating the diffusion to the local entropy, we have shown that there is an increase in local entropy in the crystal state, which is suggested to be the driving force for colloidal crystallization. As the concentration is increased to above the glass transition, a very similar characteristics as in the liquid is found. The corresponding S(q) also shows a similarity between the glass and the liquid even though S(qmax) increases to above 4 in the glass state. It is even possible to represent S(q) in the glass state with MPBRMSA calculated using the same parameters as in the liquid state. In the crystal state, it is not possible to define S(q) in the same manner as in the two other states since it is now orientation-dependent. We have shown that the liquid-crystal transition can be identified in addition to the appearance of Bragg peaks with a short-time dynamic transition since ds is speeding up. On the other hand, no sharp transition in the short-time dynamics or S(q) can be found between the glass and the liquid. It is only the freezing and consequently the nonergodisity that distinguish the dynamic transition between the liquid and the glass.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank M. P. Letting for fruitful and helpful discussions and G. Nägele and M. Heinen for providing the code for PBRMSA and D(q) calculations.
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REFERENCES
(1) Pusey, P. N. In Liquids, Freezing and Glass Transition; Levesque, D., Hansen, J. P., Zinn-Justin, J., Eds.; Elsevier: Amsterdam, 1991. (2) Asakura, S.; Oosawa, F. On Interaction between 2 Bodies Immersed in a Solution of Macromolecules. J. Chem. Phys. 1954, 22, 1255−1256. (3) de Hek, H.; Vrij, A. Interactions in Mixtures of Colloidal Silica Spheres and Polystyrene Molecules in Cyclohexane 0.1. Phase Separations. J. Colloid Interface Sci. 1981, 84, 409−422. (4) Kleshchanok, D.; Tuinier, R.; Lang, P. R. Depletion interaction mediated by a polydisperse polymer studied with total internal reflection microscopy. Langmuir 2006, 22, 9121−9128. (5) Vrij, A. Polymers at Interfaces and Interactions in Colloidal Dispersions. Pure Appl. Chem. 1976, 48, 471−483. (6) Tadros, T. Electrostatic and Steric Stabilization of Colloidal Dispersions; John Wiley & Sons: Hoboken, NJ, 2012. (7) Banchio, A. J.; Gapinski, J.; Patkowski, A.; Haussler, W.; Fluerasu, A.; Sacanna, S.; Holmqvist, P.; Meier, G.; Lettinga, M. P.; Nägele, G. Many-body hydrodynamic interactions in charge-stabilized suspensions. Phys. Rev. Lett. 2006, 96, 138303. 6682
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(33) Pedersen, J. S. Analysis of small-angle scattering data from colloids and polymer solutions: modeling and least-squares fitting. Adv. Colloid Interface Sci. 1997, 70, 171−210. (34) Beenakker, C. W. J.; Mazur, P. Diffusion of Spheres in a Concentrated Suspension 0.2. Physica A 1984, 126, 349−370. (35) Cichocki, B.; Ekiel-Jezewska, M. L.; Wajnryb, E. Lubrication corrections for three-particle contribution to short-time self-diffusion coefficients in colloidal dispersions. J. Chem. Phys. 1999, 111, 3265− 3273. (36) Pusey, P. N. Freezing and melting: Action at grain boundaries. Science 2005, 309, 1198−1199. (37) Ackerson, B. J. Entropy - When Order Is Disordered. Nature 1993, 365, 11−12.
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