ARTICLE pubs.acs.org/JPCB
Clusters in Colloidal Systems Annalisa Fierro,* Tiziana Abete, Antonio Coniglio, and Antonio de Candia CNR-SPIN and Department of Physics, University of Naples, via Cinthia, 80126 Napoli, Italy INFN, Unit of Naples, Napoli, Italy ABSTRACT: We study the dynamical properties of a model for charged colloidal particles, performing molecular dynamics simulations and observing the behavior of bond persistence functions, self-intermediate scattering functions at different wave vectors, and mean-square displacements of the particles, in three different regimes of the volume fraction. At the lowest volume fraction the system displays properties very similar to those of a gelling system, which can be interpreted in terms of the distribution of cluster sizes, with a peak in the dynamical susceptibility at the lowest wave vector. At the highest volume fraction, a percolating network of bonds is always present, and the system is strongly reminiscent of strong glasses, with the maximum in the dynamical susceptibility increasing when the temperature is lowered, and an Arrhenius dependence of the relaxation times. At intermediate volume fractions, a complex behavior is found, where both the distribution of cluster sizes and the intercluster correlations due to crowding are important.
’ INTRODUCTION In the present paper we study a Derjaguin, Landau, Verwey and Overbeek (DLVO)-type model1 for charged colloidal systems.2-4 Due to the competition between short-range attraction and long-range electrostatic repulsion, the system presents a very complex phase diagram. At low temperatures, increasing the volume fraction, the system undergoes a transition from a disordered cluster phase to an ordered hexagonal lattice of tubular structures and, at higher volume fraction, to an ordered lamellar phase.5 If these ordered states are avoided, the system enters a “supercooled” metastable liquid phase until structural arrest with gel features occurs6,7 very close to the percolation threshold. At low volume fraction, the dynamical heterogeneities can be well described in terms of clusters made of “mobile” particles, connected by “persistent” bonds: the mean cluster size of such clusters reproduces the observed behavior of the dynamical susceptibility.8-10 At higher volume fractions, the dynamical susceptibility shows a discrepancy with such mean cluster size, indicating a crossover toward a new regime, where, in addition to the clusters, the crowding of the particles also starts to play a role in the slowing down of the dynamics.11 In this paper, we study the formation and the diffusion of clusters, upon decreasing the temperature, at three values of the volume fraction, chosen in the regions of low, high, and intermediate volume fraction. At low volume fraction the behavior of the system can be described in terms of diffusing clusters. In this regime the relaxation time of the bonds between particles is much greater than the times characteristic of the diffusion of the clusters. At r 2011 American Chemical Society
high volume fraction, the system shows glassy dynamics upon lowering the temperature, resembling that of strong glasses. In this case, the particles are blocked in cages formed by the bonds with nearest neighbors, so that the two times are strongly connected. In the intermediate regime, we observe a crossover from the gel-like to the glassy-like regime, where the slow dynamics and the structural arrest display very peculiar features.
’ MATERIALS AND METHODS We consider a DLVO-type potential1 for a charged colloidal system. In our system, two particles i and j interact via the potential " # 6 σij 36 σij e-r=ξ Vij ðrÞ ¼ ε A -B þC ð1Þ r r r=ξ where r is the distance between particles, A = 3.56, B = 7.67, C = 75.08, ξ = 0.49, and σij = (σi þ σj)/2. A small degree of polydispersity is introduced in order to hinder the formation of the ordered phases.5 In particular, the radii σi are randomly distributed in the interval σ - δ/2 < σi < σ þ δ/2 with δ = 0.05σ. The potential is truncated and shifted to zero at a distance of 3.5σ. The temperature T is in units of ε/kB, where kB is the Boltzmann constant, and times in units of (mσ2/ε)1/2, where m is Special Issue: Clusters in Complex Fluids Received: October 20, 2010 Revised: January 13, 2011 Published: February 14, 2011 7281
dx.doi.org/10.1021/jp110048h | J. Phys. Chem. B 2011, 115, 7281–7287
The Journal of Physical Chemistry B
ARTICLE
Figure 1. (a) Number of particles belonging to clusters of size s, sn(s), for φ = 0.07 and T = 2 (black circles), 0.7 (red triangles), 0.5 (blue stars), 0.3 (red circles), and 0.15 (blue triangles). (b) sn(s) for φ = 0.12 and T = 0.5 (black circles), 0.4 (red triangles), 0.3 (blue stars), 0.25 (red circles), and 0.15 (black triangles). Lines are guides for the eyes.
the mass of the particles. The volume of the simulation box is kept constant, V = 5000πσ3/3, and different volume fractions are obtained by varying the number of particles N, so that φ = πσ3N/ 6L3. We perform Newtonian molecular dynamics at constant NVT using the velocity Verlet algorithm and the Nose-Hoover thermostat12 with time step δt = 0.01t0 (where t0 = (mσ2/ε)1/2 and m is the mass of the particles). The case of low temperature (T = 0.15) was studied in previous works,10,11 where it has been shown that at low volume fraction the competition between attraction and repulsion leads to the formation of a cluster phase with stable clusters of finite size. Upon increasing the volume fraction, such clusters aggregate into helical elongated structures, which eventually percolate into a disordered gel network (gel-like regime). Further increasing the volume fraction, in addition to the clusters, the crowding of the particles also starts to play a role in the slowing down of the dynamics (glassy-like regime). Here we study the formation and the diffusion of clusters upon decreasing the temperature at three values of the volume fraction, respectively, in the gel-like and in the glassy-like regime (φ = 0.07, 0.18), and in the crossover region (φ = 0.12), where the structural arrest displays very peculiar features.11
’ RESULTS AND DISCUSSION A detailed study of the percolation threshold by varying the temperature is in progress. For our scope it is enough to know that the percolation threshold is a decreasing function of the temperature: the volume fraction φ = 0.12 is above the threshold for T > 0.3 and below the threshold for T e 0.3; φ = 0.07 is below the threshold for all temperatures here considered, and φ = 0.18 is always above. In Figure 1a,b, the number of particles belonging to clusters of size s, sn(s) (where n(s) is the cluster size distribution) is plotted at different temperatures, respectively, for φ = 0.07 and φ = 0.12. We find that at high temperature, particles behave as hard spheres and are randomly distributed; decreasing the temperature, as a result the attractive interaction, particles aggregate to form clusters; further decreasing the temperature, due to the competition between short-range attraction and long-range repulsion, the structure of the clusters modifies, and a cluster phase with finite size appears. This region
of the phase diagram is characterized by two well-separated time scales, the lifetime of bonds between particles, τb, and the structural relaxation time, τR, which differ by several orders of magnitude, with τbττR.10,11 Hence, for times less than the bond lifetime, clusters are well-defined and diffuse as large particles. The bond lifetime,10 τb, is defined by B(τb) = 0.1, where BðtÞ ¼
Σij ½Ænij ðtÞnij ð0Þæ - Ænij æ2 Σij ½Ænij æ - Ænij æ2
ð2Þ
is the time autocorrelation function of bonds. The variable nij(t) is equal to 1 if particles i and j are bonded at time t, and is equal to zero otherwise.13 In eq 2, Æ...æ is the thermal average and [...] is the average over different realizations of the system (in particular, here we consider 32 independent realizations of the system for each choice of the control parameters). In Figure 2a, B(t) is plotted for φ = 0.07 and different temperatures. The bond lifetime obtained in this way increases as a function of the inverse temperature as an Arrhenius law B exp(A/T) (straight lines in Figures 2b and 3a,b). Self-Intermediate Scattering Function and Dynamical Susceptibility. In refs 10 and 11, the self-intermediate scattering function (ISF) and the dynamical susceptibility were studied in detail at low temperature and low volume fraction. In this section we first recall the case of low volume fractions (φ = 0.07, 0.12) and then consider the case of φ = 0.18. We measure the selfISF, defined as 1 ½ÆΦs ðk, tÞæ N " # N 1 X i Bk Æ e 3ð B r i ð0ÞÞæ r i ðtÞ - B ¼ N i¼1
Fs ðk, tÞ ¼
ð3Þ
and the dynamical susceptibility χ4 ðk, tÞ ¼
1 ½ÆjΦs ðk, tÞj2 æ - jÆΦs ðk, tÞæj2 N
ð4Þ
The structural relaxation time, τR(k), is obtained as Fs(k,τR(k)) = 0.1. In Figures 2b and 3a,b, τR(k) is plotted respectively for φ = 0.07, 0.12, and 0.18 and several wave vectors. We find that 7282
dx.doi.org/10.1021/jp110048h |J. Phys. Chem. B 2011, 115, 7281–7287
The Journal of Physical Chemistry B
ARTICLE
Figure 2. (a) Time autocorrelation functions of bonds for φ = 0.07 and T = 0.7, 0.65, 0.5, 0.4, 0.3, 0.2, 0.175, and 0.15 (from left to right). (b) Structural relaxation time, τR(k), at k = 0.36, 0.72, 1.88, (from top to bottom) compared with the bond lifetime (black triangles) for φ = 0.07 as a function of the inverse temperature, 1/T. The straight line is an Arrhenius law, τb ∼ B exp(A/T).
Figure 3. (a) Structural relaxation time, τR(k), at k = 0.36, 0.72, and 1.88, (from top to bottom) compared with the bond lifetime τb (black triangles) for φ = 0.12 as a function of the inverse temperature, 1/T. Straight lines are Arrhenius laws, τ ∼ B exp(A/T). (b) Structural relaxation time, τR(k), at k = 6.87, compared with the bond lifetime τb (black triangles) for φ = 0.18 as a function of the inverse temperature, 1/T. Straight lines are Arrhenius laws, τ ∼ B exp(A/T).
τR(k) displays the same dependence on T by changing k. In particular, τR(k) increases weakly as a function of the inverse temperature for φ = 0.07, whereas for both φ = 0.12 and φ = 0.18 it follows an Arrhenius law. In Figure 4a,b, Fs(k,t) and χ4(k,t) are plotted for the minimum value of the wave vector, kmin = 0.36, φ = 0.07, and different temperatures. At high temperature, χ4(k,t) displays a small peak and sharply tends to its asymptotic value, 1. Decreasing the temperature, the peak increases until a large plateau develops at intermediate times, as already found in refs 10 and 11. This behavior strongly resembles that found in permanent gels,14 where the dynamical susceptibility at low wave vector tends to a plateau coinciding with the mean cluster size, and shows that, in this region of the phase diagram, the system behaves as a gelling system. Increasing the volume fraction, a different behavior is observed signaling the crossover to a different regime. For φ = 0.12, no plateau is present at intermediate times, and a well-pronounced peak is always found, as we can see in Figure 5b, where χ4(k,t) is plotted. In Figure 5a, the corresponding self-ISF Fs(k,t) is shown.
In ref 10 it has been shown that at low temperature and low volume fraction the dynamical susceptibility in the limit of zero k is reproduced by the mean cluster size of clusters made of “mobile’’ particles connected by “persistent” bonds. Increasing the volume fraction, at φ = 0.12, the dynamical susceptibility shows a discrepancy with the time-dependent mean cluster size, indicating a crossover toward a new regime where, besides the clusters, also the crowding of the particles starts to play a role. Moreover,11 at this volume fraction, the maximum of the dynamical susceptibility χ4(k,t*) is not a monotonic decreasing function of k (as instead at lower φ), whereas it displays a maximum for ~k = 0.72. Further increasing the volume fraction, a similar picture is found at intermediate temperature. In Figure 6a,b, Fs(k,t) and χ4(k,t) are respectively plotted, for φ = 0.18, T = 0.25, and different wave vectors. As we see, χ4(k,t*) displays a maximum for ~k = 4.09. We find that ~k increases by decreasing temperature until, at the lower temperature considered here, a weaker dependence on the wave vector is found in χ4(k,t*), which 7283
dx.doi.org/10.1021/jp110048h |J. Phys. Chem. B 2011, 115, 7281–7287
The Journal of Physical Chemistry B
ARTICLE
Figure 4. (a) The self-ISF Fs(k,t), for wave vector k = 0.36 at φ = 0.07 and T = 0.65, 0.4, 0.3, 0.2, and 0.15 (from left to right). (b) The dynamical susceptibility χ4(k,t), for k = 0.36 at φ = 0.07 and T = 0.65, 0.4, 0.3, 0.2, and 0.15 (from bottom to top to right).
Figure 5. (a) The self-ISF Fs(k,t), for wave vector k = 0.36 at φ = 0.12 and T = 0.5, 0.4, 0.3, 0.2, and 0.15 (from left to right). (b) The dynamical susceptibility χ4(k,t), for k = 0.36 at φ = 0.12 and T = 0.5, 0.4, 0.3, 0.2, and 0.15 (from bottom to top).
displays a large maximum around ~k = 7, as observed in glassy systems.15 In Figure 7a,b, Fs(k,t) and χ4(k,t) are respectively plotted for φ = 0.18, k = 6.87, and different temperatures. At low temperature, large oscillations appear both in Fs(k,t) and χ4(k,t) at the crossover from the ballistic to the caging regime (see also Figure 9a,b), reminiscent of the Boson peak found in BKS silica.16 In conclusion, for φ = 0.18, the system displays dynamical properties strongly recalling those of strong glasses. Mean-Square Displacement and Non-Gaussian Parameters. In order to study the diffusion of long living clusters and their connection with dynamics heterogeneities, we measure the mean square displacement (MSD) ÆΔr 2 ðtÞæ ¼
N 1X ½Æj r ðtÞ - B r i ð0Þj2 æ N i ¼ 1 Bi
and the non-Gaussian parameter:17 " # 3ÆΔr 4 ðtÞæ R2 ðtÞ ¼ -1 5ÆΔr 2 ðtÞæ2
ð5Þ
ð6Þ
where Br i(t) is thePposition of the ith particle at the time t, and 4 Δr4(t) = (1/N) N i=1[Æ|r Bi(t) - Br i(0)| æ]. R2(t) is zero if the probability distribution of the particle displacements (the selfpart of the Van-Hove function) is Gaussian. The MSD is plotted for φ = 0.07, 0.12, and 0.18, respectively, in Figures 8a,b, and Figure 9a. At high temperature, the system behaves as a simple fluid. We find that ÆΔr2(t)æ is well fitted by 6D(t - (1 - e-ξt)/ξ) (the upper curves plotted in Figure 8a for φ = 0.07 and T = 1.5, 0.65) as expected for Brownian particles satisfying the Langevin equation,18 where D is the diffusion coefficient, ξ is the friction coefficient, and ξD = KBTm-1. Correspondingly, R2(t) displays a small maximum almost at the crossover from the ballistic (ÆΔr2(t)æ t2) to the diffusive regime (ÆΔr2(t)æ t). Decreasing the temperature, deviations from this behavior are found. In particular, the ballistic regime is observed at very short times (smaller than 0.1), followed by a region of anomalous superdiffusion (ÆΔr2(t)æ tx, with 1 < x < 2), and finally by the diffusive regime. Correspondingly, a small peak develops at short times, roughly in coincidence with the superdiffusive regime. We speculate that the superdiffusive regime and the short time peak 7284
dx.doi.org/10.1021/jp110048h |J. Phys. Chem. B 2011, 115, 7281–7287
The Journal of Physical Chemistry B
ARTICLE
Figure 6. (a) The self-ISF Fs(k,t), for φ = 0.18, T = 0.25, and wave vector k = 0.36, 1.08, 2.05, 3.07, 4.09, 5.42, 6.87, 7.52, 8.32, and 9.40 (from right to left). (b) The dynamical susceptibility χ4(k,t), for φ = 0.18, T = 0.25, and wave vector k = 0.36, 1.08, 2.05, 3.07, 4.09, 5.42, 6.87, 7.52, 8.32, and 9.40 (from right to left). Lines are guides for the eyes.
Figure 7. (a) The self-ISF Fs(k,t), for wave vector k = 6.87 at φ = 0.18 and T = 0.45, 0.35, 0.3, 0.25, 0.2, 0.175, and 0.15 (from bottom to top). (b) The dynamical susceptibility χ4(k,t), for k = 6.87 at φ = 0.18 and T = 0.45, 0.35, 0.3, 0.25, 0.2, and 0.175 (from bottom to top).
Figure 8. (a) MSD for φ = 0.07 and T = 1.5, 0.65, 0.4, 0.3, and 0.15 (from top to bottom). The two solid lines are from top to bottom: 6D(t - (1 - eξt )/ξ), with (1) 6D = 3.62 and ξ = 1.00 and (2) 6D = 8.36 and ξ = 1.05. (b) MSD for φ = 0.12 and T = 0.5, 0.4, 0.3, 0.2, 0.15, and 0.125 (from top to bottom). The three solid lines are from top to bottom: atx with (1) a = 0.70 and x = 0.4, (2) a = 0.70 and x = 0.26, and (3) a = 0.70 and x = 0.18. 7285
dx.doi.org/10.1021/jp110048h |J. Phys. Chem. B 2011, 115, 7281–7287
The Journal of Physical Chemistry B
ARTICLE
Figure 9. (a) MSD for φ = 0.18 and T = 0.5, 0.45, 0.4, 0.35, 0.3, 0.25, 0.2, 0.175, and 0.15 (from top to bottom). The three solid lines are, from top to bottom, atx with (1) a = 0.25 and x = 0.65, (2) a = 0.16 and x = 0.45, (3) a = 0.055 and x = 0.31, (4) a = 0.055 and x = 0.16, and (5) a = 0.055 and x = 0.08. (b) Non-Gaussian parameter for φ = 0.18 and T = 0.25, 0.2, 0.175, and 0.15 (from bottom to top).
are both due to the presence of clusters. In order to verify this hypothesis, we also measure the MSD of clusters with fixed size,19 ÆΔr2(s,t)æ, and the corresponding non-Gaussian parameter, R2(s, t). We find that the superdiffusive regime and the peak are not present for single particles. In particular, the short time peak develops for size s > 1 and increases as a function of s. Further decreasing the temperature, for φ = 0.12 and φ = 0.18, a subdiffusive regime (ÆΔr2(t)æ tx, with x < 1) develops at intermediate time, before the final crossover to the diffusive regime. This “caging” subdiffusive region is more and more extended as the temperature decreases and the volume fraction increases. The exponent x characterizing the subdiffusive regime is an increasing function of the temperature.20 Correspondingly, the non-Gaussian parameter R2(t) displays a peak, in coincidence with the crossover from the caging region to the diffusive one, which increases by decreasing the temperature (see Figure 9b).
’ CONCLUSIONS Using molecular dynamics simulations, we study a DLVOtype model for charged colloidal systems. We show that in this system dynamical properties are strongly linked to cluster distribution. At low volume fraction, φ = 0.07, long-living clusters of finite size appear by decreasing temperature, and dynamical properties of a gelling system are found: one-step decay in the self-ISF; no caging regime in the MSD; plateau in the dynamical susceptibility, coinciding at low wave vector with the mean cluster size of mobile particles connected by persistent bonds; and structural relaxation time weakly increasing by decreasing the temperature. At high volume fraction, φ = 0.18, almost all particles belong to the same spanning cluster. By decreasing temperature, properties strongly recalling those of strong glasses appear: two-step decay in the self ISF; a caging regime in the MSD; large oscillations at the crossover from the ballistic regime to the caging one, reminiscent of the Boson peak; dynamical susceptibility and non-Gaussian parameter with the maximum increasing as the temperature decreases; and Arrhenius dependence of the structural relaxation time on temperature. Here the cage, which is not
due to crowding, is instead due to bonds linking nearest neighbors particles, as in attractive glasses. At intermediate volume fraction, φ = 0.12, near the percolation threshold, the system displays properties in between. On one hand we observe caging regime in the MSD; dynamical susceptibility and non-Gaussian parameter with the peak increasing as temperature decreases; and Arrhenius dependence of the structural relaxation time on temperature. On the other hand, the behavior of the self-ISF may be interpreted as superposition of different cluster contributions,11,21 as in permanent gels. The dynamical susceptibility at low wave vector, which shows a peak not representable in terms of clusters, coincides with the timedependent mean cluster size10 in both short and long time regions. These findings indicate a crossover toward a new regime where, besides the clusters, other mechanisms contribute to dynamical heterogeneities.
’ AUTHOR INFORMATION Corresponding Author
*E-mail: fi
[email protected].
’ ACKNOWLEDGMENT This work is partially supported by the CNR-INFM Parallel Computing Initiative and the Italian SuperComputing Resource Allocation - ISCRA. ’ REFERENCES (1) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1985. Crocker, J. C.; Grier, D. G. Microscopic Measurement of the Pair Interaction Potential of Charge-Stabilized Colloid. Phys. Rev. Lett. 1994, 73, 352–355. (2) Dinsmore, A. D.; Prasad, V.; Wong, I. Y.; Weitz, D. A. Microscopic Structure and Elasticity of Weakly Aggregated Colloidal Gel. Phys. Rev. Lett. 2006, 96, 185502. (3) Puertas, A. M.; Fuchs, M.; Cates, M. E. Mode Coupling and Dynamical Heterogeneity in Colloidal Gelation: A Simulation Study. J. Phys. Chem. B 2005, 109, 6666–6675. (4) Sciortino, F; Tartaglia, P.; Zaccarelli, E. One-Dimensional Cluster Growth and Branching Gels in Colloidal Systems with Short-Range 7286
dx.doi.org/10.1021/jp110048h |J. Phys. Chem. B 2011, 115, 7281–7287
The Journal of Physical Chemistry B
ARTICLE
Depletion Attraction and Screened Electrostatic Repulsion. J. Phys. Chem. B 2005, 109, 21942–21953. (5) de Candia, A.; Del Gado, E.; Fierro, A.; Sator, N.; Tarzia, M.; Coniglio, A. Columnar and Lamellar Phases in Attractive Colloidal Systems. Phys. Rev. E 2006, 74, 010403. (6) Coniglio, A.; De Arcangelis, L.; Del Gado, E.; Fierro, A.; Sator, N. Percolation, Gelation and Dynamical Behaviour in Colloids. J. Phys: Condens. Matter 2004, 16, S4831–S4839. de Candia, A.; Del Gado, E.; Fierro, A.; Sator, N.; Coniglio, A. Colloidal Gelation, Percolation and Structural Arrest. Physica A 2005, 358, 239–248. (7) Tarzia, M.; Coniglio, A. Pattern Formation and Glassy Phase in the Φ4 Theory with a Screened Electrostatic Repulsion. Phys. Rev. Lett. 2006, 96, 075702. (8) Franz, S.; Parisi, G. On Non-linear Susceptibility in Supercooled Liquids. J. Phys: Condens. Matter 2000, 12, 6335–6342. (9) Donati, C.; Franz, S.; Glotzer, S. C.; Parisi, G. Theory of Nonlinear Susceptibility and Correlation Length in Glasses and Liquids. J. Non-Cryst. Solids 2002, 307, 215–224. (10) Fierro, A.; de Candia, A.; Del Gado, E.; Coniglio, A. Dynamic Heterogeneities in Attractive Colloids. J. Stat. Mech. 2008, L04002. (11) de Candia, A.; Del Gado, E.; Fierro, A.; Coniglio, A. Length Scale Dependenceof the Dynamical Heterogeneities in Colloidal Gelation at Low Volume Fraction. J. Stat. Mech. 2009, P02052. (12) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford University Press: Oxford, 1987. (13) Two particles are considered linked if their relative distance is smaller than the local maximum of the potential. (14) Abete, T.; de Candia, A.; Del Gado, E.; Fierro, A.; Coniglio, A. Static and Dynamic Heterogeneities in a Model for Irreversible Gelation. Phys. Rev. Lett. 2007, 98, 088301. Abete, T.; de Candia, A.; Del Gado, E.; Fierro, A.; Coniglio, A. Dynamical Heterogeneity in a Model for Permanent Gels: Different Behavior of Dynamical Susceptibilities. Phys. Rev. E 2008, 78, 041404. (15) Charbonneau, P.; Reichman, D. R. Dynamical Heterogeneity and Nonlinear Susceptibility in Supercooled Liquids with Short-Range Attraction. Phys. Rev. Lett. 2007, 99, 135701. (16) Horbach, J.; Kob, W.; Binder, K. The Dynamics of Supercooled Silica: Acoustic Modes and Boson Peak. J. Non-Cryst. Solids 1998, 235, 320–324. (17) Rahman, A. Correlations in the Motion of Atoms in Liquid Argon. Phys. Rev. 1964, 136, A405–A411. (18) Hansen, J. -P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic: London, 1986. (19) We measure the MSD of particles that at time t = 0 belong to a given cluster of size s. Clearly at high temperature the MSD of particles that at time t = 0 belong to a cluster of size s is nondependent on s and coincides with the average MSD, as expected for very low bond lifetimes. Decreasing the temperature from 0.65 to 0.15, the bond lifetime increases by many orders of magnitude (see Figures 2b and 3a,b), and at the lower values of the temperature considered here, clusters live for the whole observation time window. (20) For φ = 0.18, x(T) can be fitted by the linear function b(T T0) with b = 3.83 and T0 = 0.13. (21) Fierro, A.; Abete, T.; Coniglio, A. Emergence of Complex Behavior in Gelling Systems Starting from Simple Behavior of Single Clusters. J. Chem. Phys. 2009, 131, 194906.
7287
dx.doi.org/10.1021/jp110048h |J. Phys. Chem. B 2011, 115, 7281–7287
