Anomalous dynamics of binary colloidal mixtures over a potential barrier: Effect of depletion interaction A. V. Anil Kumar Citation: The Journal of Chemical Physics 141, 034904 (2014); doi: 10.1063/1.4890282 View online: http://dx.doi.org/10.1063/1.4890282 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Determination of favorable inter-particle interactions for formation of substitutionally ordered solid phases from a binary mixture of oppositely charged colloidal suspensions J. Chem. Phys. 138, 174504 (2013); 10.1063/1.4802784 Scaling between relaxation, transport and caged dynamics in a binary mixture on a per-component basis J. Chem. Phys. 138, 12A532 (2013); 10.1063/1.4789943 Structure, compressibility factor, and dynamics of highly size-asymmetric binary hard-disk liquids J. Chem. Phys. 137, 104509 (2012); 10.1063/1.4751546 Diffusion in a nonequilibrium binary mixture of hard spheres swelling at different rates J. Chem. Phys. 131, 024503 (2009); 10.1063/1.3168405 Influence of a depletion interaction on dynamical heterogeneity in a dense quasi-two-dimensional colloid liquid J. Chem. Phys. 121, 8627 (2004); 10.1063/1.1800951
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THE JOURNAL OF CHEMICAL PHYSICS 141, 034904 (2014)
Anomalous dynamics of binary colloidal mixtures over a potential barrier: Effect of depletion interaction A. V. Anil Kumara) School of Physical Sciences, National Institute of Science Education and Research, IOP Campus, Bhubaneswar 751005, India
(Received 14 May 2014; accepted 3 July 2014; published online 21 July 2014) The dynamics of a binary colloidal mixture under the influence of an external potential barrier has been studied by molecular dynamics simulations. The attractive depletion interaction between the barrier and larger particles fastens the dynamics of the larger particles over the potential barrier. At low temperatures, depletion interactions cause the larger particles to diffuse faster than smaller particles, which is counterintuitive. The repulsive barrier leads the small particles to undergo an anomalous diffusion which resembles the dynamics of systems undergoing a glass transition, while the larger particles undergo normal diffusion even at very low temperature. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4890282] I. INTRODUCTION
Colloidal dispersions consist of mesoscopic particles dispersed in a medium of microscopic particles and are typically characterized by the dynamics of the dispersed particles.1 There are easy ways of tuning the interaction potential between the colloidal particles, which make them act as model systems to study the fundamental questions on the properties of condensed matter systems such as equilibrium and nonequilibrium phase transitions. The colloidal systems themselves exhibit a number of interesting physical phenomena such as repulsive and attractive glasses,2, 3 cluster fluids,4 shear thinning and thickening transitions,5, 6 etc. Many of these phenomena are caused by the indirect interactions between the colloidal particles. Depletion interaction is the prominent indirect interaction, which arises when particles of different sizes are present in a colloidal dispersion.7 In a suspension of large and small particles, there will be a restricted volume region surrounding the big particles where small particles cannot enter. Overlap of these restricted volumes increases the volume accessible to small particles, which causes an effective attraction between the larger particles. The range and depth of this attractive interaction depends on the size and density of the small particles. This purely entropic interaction was first proposed by Asakura and Oosawa8 for a colloid-polymer mixture and later elaborated by Vrij.9 These interactions play a crucial role in the current understanding of the phase behavior and dynamics of dispersions containing colloids of disparate sizes and has been called upon to account for many phenomena such as size and shape selectivity,10, 11 protein crystallization,12 macromolecular crowding,13, 14 and dynamics in crowded medium.15, 16 In a binary colloidal mixture, subjected to an external potential, depletion interactions not only arise due to the overlap of excluded volumes of larger particles, but also due to the overlap of excluded volumes of larger particles and that of a repulsive potential. It has been shown that when a cola) Electronic mail: [email protected]
0021-9606/2014/141(3)/034904/6/$30.00
loidal binary mixture comprising particles of disparate sizes has been confined between the hard walls, the larger particles get attracted towards the wall resulting in an increased density of them near the wall.17–19 This can be explained on the basis of depletion interactions, according to which the hard wall also possesses a depletion layer surrounding it, in which the smaller particles cannot enter. Therefore, the larger particles get attracted towards the wall due to the overlap of the excluded volumes of the wall and the larger particles. This phenomena is also true for walls with a softer, but infinite confining potential. These observations led to the suggestion that whenever a binary mixture is subjected to an external repulsive potential, there will be an effective attraction for larger particles towards the potential due to depletion interactions. This conjecture has been verified by dynamical density functional theory calculations (DDFT)20 as well as Monte Carlo simulations.21 These investigations have revealed that because of the attractive depletion interactions, the density of large particles increases in the region of external repulsive potential and the strength of this depletion interaction depends on various parameters like density and number ratio of the mixture, size ratio of the particles, and width and height of the potential barrier. However, these studies were restricted to the equilibrium properties of the binary mixture. In this article, we investigate the dynamical properties of a colloidal binary mixture subjected to an external repulsive barrier, using molecular dynamics simulations. Our results reveal that the dynamical properties of this mixture are as rich and significant as the equilibrium properties. The depletion interactions fasten the dynamics of larger particles in the binary mixture and make them diffuse faster than the smaller particles as the temperature is lowered which is counterintuitive. The smaller particles undergo a typical glass transition scenario at lower temperatures even though the density is quite low. II. THE MODEL AND SIMULATION DETAILS
We performed canonical ensemble molecular dynamics simulations of a binary mixture of purely repulsive soft 141, 034904-1
© 2014 AIP Publishing LLC
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spheres, with interaction potential Vij (rij ) given by
III. RESULTS AND DISCUSSION
To investigate the effect of depletion interactions due to the presence of external potential barrier on the dynamics of binary mixture, we have carried out a number of molecular dynamics simulations at a fixed total volume fraction φ = 0.20, varying the volume fraction of the individual components. We started with a single component fluid of larger particles and in the subsequent simulations increased the volume fraction of smaller particles keeping the total volume fraction constant. The self-diffusion coefficients were calculated from the mean squared displacement of the particles. Figure 1(a) shows the self-diffusion coefficients of larger and smaller particles against the volume fraction of smaller particles, φ s . Dzl (Dzs ) is the self-diffusion coefficient of the larger (smaller) particles along the z-direction where the particles have to encounter the potential barrier, while D0l (D0s ) is the average self-diffusion coefficients along x and y directions where there is no external potential. For the smaller particles, the self-diffusion coefficient across the barrier is always lower than the self-diffusion coefficients along other two directions, which is expected since the particles have to cross the potential barrier along z-direction. However, for the larger particles the scenario is completely different. In the case of single component fluid of larger particles, again Dzl is lesser than D0l . As
s
D0
s
Dz
D
where i, j = l, s. We chose σ ss = 1.0, σ ll = 2.0, ss = 1.0, and ll = 4.0.21 All the parameters are expressed in dimensionless reduced units. Cross interaction parameters are obtained by Lorentz-Berthelot additive mixture rules. We kept all the masses equal, ms = ml = 1.0. The mixture is subjected to a potential barrier at the center of the simulation box along the z-axis, which is of Gaussian form, z − z 2 0 . (2) Vext (z) = εext exp − w Here, w is the range of external potential and εext is the height of the potential. We used w = 3.0 and εext = 2.0. Recent advances in experimental techniques make it possible to create such localized potential barriers.22 The simulations solve the system of first order differential equations q˙ i = pi /m, p˙i = Fi − λpi , where λ is the thermostat factor determined by the Gaussian principle of least constraint,23 qi and pi are the position and momentum vectors, respectively, and Fi is the net force acting on the particle i. The above equations are solved simultaneously using a fifth order Gear predictorcorrector method.24 We used a cubic simulation box of length Lb = 20.48 and periodic boundary conditions are imposed along all the three directions. We used a time step dt = 0.001 in reduced units (t0 = [ms σss2 /ss ]1/2 ). In each simulation run, we allowed the system to equilibrate for 5× 106 time steps and then the measured properties are averaged over a production run of 20× 106 steps. The dynamical properties are calculated from the trajectories produced in each production run. These properties are averaged over four simulation runs starting from different initial configurations.
l
Dz
1
(1)
0.5
0
0
0.05
φs
0.1
large small
(b)
1 Dz/D0
Vij (rij ) = εij (σij /rij )36 ,
l
D0
(a)
0.8 0.6 0.4 0.2
0
0.05
φs
0.1
FIG. 1. (a) Self-diffusion coefficients of small and large particles over the potential barrier and parallel to the potential barrier against the volume fraction, φ s of small particles. The larger particle volume fraction is φ − φ s . Here, φ is fixed at 0.20. Dzl and Dzs are the self-diffusivities of large and small particles along the z-direction. D0l and D0s are the self-diffusivities in the x − y plane. (b) The ratio of self-diffusion coefficients over the barrier to that parallel to the barrier for both small and large particles.
we increase the volume fraction of the smaller particles, the difference between D0l and Dzl starts decreasing and they become almost similar when we have a significant volume fraction smaller particles present in the mixture. The introduction of smaller particles into the system invokes the depletion interaction among the larger particles as well as among the large particles and the potential barrier. So the large particles feel an attraction towards the barrier which will enable them to cross the barrier easily. This increases Dzl and the larger particles diffuse as if they do not feel the presence of the barrier at all. This can be easily seen from Figure 1(b), where we have plotted the ratio Dzs /D0s and Dzl /D0l against φ s . Dzs /D0s remains much lower than one for all values of φ s which is as expected. However, Dzl /D0l starts at value of 0.75 and increases as φ s increases and becomes close to one at φ s = 0.10. At even higher volume fractions of smaller particles, i.e., above φ s = 0.10, the ratio starts decreasing. Earlier Monte Carlo simulations have shown that the depletion interaction for this system increases as φ s increases, goes through a maximum at φ s = 0.10 and then decreases.21 The results of molecular dynamics simulations agree with these observations. We concentrated on an equi-volume mixture with total volume fraction, φ = 0.20 for further investigations, since the depletion interaction is a maximum for this mixture. To investigate the effect of the potential barrier on the dynamics of the particles, we calculated the self-diffusion-coefficients along the potential barrier and perpendicular to it. Figure 2 depicts the temperature variation of the self-diffusion coefficients, Dzl , D0l , Dzs , and D0s . D0l and D0s show the typical temperature dependence for fluids. The smaller particles diffuse faster than the larger particles at all temperatures studied. However, the diffusion along the z-direction where the
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2.5
1
(a)
T=2.0 T=1.0 T=0.7 T=0.5 T=0.4 T=0.3 T=0.25
2 ρl(z)
0.1 z
D
Dl
Ds
0.01
1.5 1
z
D0 D0
0.5
l s
0
0.001 0
0.5
1
1.5
2
0
5
10 z
15
20
10 z
15
20
2.0
2.5
T
(b)
FIG. 2. Self-diffusivities of small and large particles along z-axis and in the x − y plane against temperature. The diffusivities along the z-direction show a crossover at T = 0.7.
ρs(z)
1.5 1.0 0.5 0 0
5
FIG. 3. Density profile as a function of z for (a) large particles and (b) small particles.
and y are uniform and do not show any deviations from the bulk density. We now turn our attention to the detailed dynamics of particles along the z-direction where they have to overcome the external potential barrier. Figure 4 displays the mean-squared displacements of both the species along the z-direction at various temperatures. At higher temperatures, smaller particles diffuse normally over the whole time and the mean-squared displacement increases linearly with time. However, at low temperatures we observe a standard glasstransition scenario: after the initial ballistic regime, there is a plateau which indicates localization of particles at the intermediate time, before diffusive region is observed at long 4
10
(a)
2
δrl
2
10
T=2.0 T=1.0 T=0.7 T=0.5 T=0.4 T=0.3 T=0.25
0
10
-2
10
-1
10
0
10
1
2
10
10
3
10
4
10
t 4
10
(b)
2
2
10 δrs
particles have to overcome the external potential barrier shows interesting temperature dependence. At high temperatures, the smaller particles diffuse faster than the bigger particles, which is expected. However, as the temperature is lowered, Dzs decreases more strongly compared to Dzl . As the temperature is lowered the smaller particles find it increasingly difficult to surmount the potential barrier and their diffusion coefficient decreases drastically. However, the larger particles feel the attraction towards the barrier due to depletion interaction and this attraction fastens their crossing over the barrier. Hence, they do not slow down appreciably and still retain the diffusive regime. Among the remarkable features of these results is that the extent of drop in the self-diffusivity of smaller particles relative to that of larger particles leads to the larger particles diffusing faster compared to smaller particles below T = 0.7. As we go down further, Dzs decreases rapidly, indicating localization of particles. The above observations suggest that the depletion interaction causes the larger and smaller particles to experience the potential barrier differently. This can be visualized by calculating the density profile of large and small particles along zdirection, ρ(z). Figure 3 shows the density profile of large and small particles at different temperatures. It is evident from the figure that the larger particles get attracted towards the barrier and smaller particles get depleted from it. As temperature decreases, the density of larger particles increases near the barrier and slowly the peak at the barrier splits in to two. The density profile of smaller particles exhibits a contrasting behavior. Within the region of the potential barrier, their density is much smaller than the bulk density and this difference increases as we decrease the temperature. The potential of mean force is given by −kB Tlnρ(z), the derivative of which with respect to z gives the effective force the barrier exerts on the particles. The figure shows clearly that the effective force on the smaller particles is increasingly repulsive as the temperature goes down. For the larger component, the force is attractive. At lower temperatures, when the density profile peak splits in to two, the potential of mean force has a small barrier at the middle, which results in a small repulsive effective force. This explains the slight reduction in the diffusivity of larger particles at low temperatures. We should also mention that the density profiles for both large and small particles along the x
0
10
-2
10
-1
10
0
10
1
2
10
10
3
10
4
10
t FIG. 4. Mean squared displacement of (a) large and (b) small particles along the z-direction.
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4
0.5
10
(a)
(a)
0.4
2
T=2.0 T=1.0 T=0.7 T=0.5 T=0.4 T=0.3 T=0.25
0
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-1
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T=2.0 T=1.0 T=0.7 T=0.5 T=0.4 T=0.3 T=0.25
0.3
l
δrl,0
2
αz (t)
10
0.2 0.1 0 -1 10
4
10
0
10
1
10
3
10
4
10
t
t 1.5
4
10
(b)
(b)
1.25
2
1.0 s
2
αz (t)
10 δrs,0
2
10
0
10
0.75 0.5 0.25
-2
10
-1
10
0
10
1
2
10
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3
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0 -1 10
4
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0
10
1
2
10
10
3
10
4
10
t
t FIG. 5. Mean squared displacement of (a) large and (b) small particles averaged along the x- and y-directions.
FIG. 6. Non-Gaussian parameter α z as a function of time for (a) large particles and (b) small particles.
times. The time window over which this plateau extends increases as the temperature decreases. For larger particles, the mean-squared displacement is linear with time in the whole temperature regime we have investigated. This should be contrasted with the average mean squared displacement along x- and y-directions, where the particles do not need to encounter any potential barrier. This is depicted in Figure 5. Mean squared displacement of both large and small particles are linear in time and does not show any signatures of slowing down. This behavior makes it clear that the potential barrier is responsible for the dynamical slowing down of smaller particles. This should have been the case for larger particles also, however, the attractive depletion interaction between the larger particle and potential barrier keeps the particle in the diffusive regime. The deviation between the measured mean squared displacement and what is expected from normal diffusive behavior can be ascertained by the non-Gaussian parameter α z (t), defined by25
termediate times it increases go through a maximum and at large times it becomes again close to zero. The intensity of the maximum in α z (t) increases with decrease in temperature. Also the peak in α z (t) moves towards higher values of t as we decrease the temperature. This behavior of α z (t) is typical of super cooled liquids near to glass transition and completely in agreement with the conclusions made based on the behavior of mean squared displacements. The above observations are further confirmed by the time evolution of z-component of intermediate scattering function Fs (k, t) . We calculated the intermediate scattering function for the motion along the z-direction as N 1 exp[ik(zj (t) − zj (0))] (4) Fs (k, t) = N j =1
αz (t) =
1 z4 − 1, 3 z2 2
(3)
where z4 and z2 are the fourth and second moment of particle displacements along the z-direction, respectively. We have calculated α z (t) for both large and small particles at various temperatures. For large particles, α z (t) is very close to zero in the entire time range we calculated, which means the particle dynamics is diffusive all the time. This is true for all the temperatures at which α z (t) is calculated. However, in the case of smaller particles we observe two scenarios with respect to the temperature (Fig. 6). At high temperature, again the non-Gaussian parameter is close to zero in the entire time range and the particle dynamics is diffusive. As we decrease the temperature to 0.7 and below, there are three distinctive regimes in α z (t). At short times, α z (t) is close to zero, at in-
with k = 2π /Lb . For larger particles, the intermediate scattering function Fs (k, t) exhibits an exponential one-step decay, which is expected for liquids, for all the temperatures studied (Fig. 7). However for smaller particles, again we observe two scenarios with respect to temperature. At high temperature, the time evolution of Fs (k, t) is a one-step exponential decay. However, as we decrease the temperature to t = 0.7 or below, a plateau develops at intermediate times, followed by a non-exponential decay to zero. Such behavior is typical of glass forming liquid. These observations are in complete agreement with the behavior of mean squared displacement and non-Gaussian parameter. There are few comments in order here. First, our results for the dynamics of smaller particles are consistent with the experimental and molecular simulation results of DalleFerrier et al.26 They have studied the dynamics of dilute colloidal suspensions in a sinusoidal potential. It has been shown that the external potential leads the particle dynamics to undergo a glass transition scenario even though the density is very small. In our simulations, we observe similar behavior
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J. Chem. Phys. 141, 034904 (2014)
ues of self-diffusion coefficients. Our results could be experimentally verified as the recent experimental advances make it possible to realise microscopically localized potential barriers using optical tweezers.22
1.0 (a) T=2.0 T=1.0 T=0.7 T=0.5 T=0.4 T=0.3 T=0.25
0.6
l
Fs (k,t)
0.8
0.4
IV. CONCLUSIONS
0.2 0 -1 10
0
10
1
2
10
10
3
4
10
10
t 1.0 (b)
0.6
s
Fs (k,t)
0.8
0.4 0.2 0 -1 10
0
10
1
2
10
10
3
10
4
10
t FIG. 7. Intermediate scattering function Fs (k, t) for the motion along the zdirection as a function of time for (a) large particles and (b) small particles. The value of k used is 2π /Lz .
for the small particle dynamics, while the large particles do not show any characteristics of dynamics of system approaching glass transition. This supports our earlier argument that the larger particle, because of the attraction due to depletion interaction do not feel the presence of the potential barrier while smaller particles feel a heightened barrier due to repulsion. Second, the comparison of our results with earlier simulation results of Voigtmann and Horbach,27 in which they considered a binary mixture of disparate sizes, without any external potential reveals an interesting point. Their simulation indicated a double transition scenario, where the larger particle undergoes a typical glass transition dynamics first, as the density is increased. Small particle dynamics will be still diffusive and undergoes a localization transition if the density is increased further. Here in our simulations, we used temperature as the control parameter instead of particle density. As we decrease the temperature, it is the smaller particle which undergoes the slowing down rather than the bigger particles. Even at the lowest temperature we simulated, i.e., T = 0.25, we have observed that the larger particles undergo normal diffusive dynamics. Thus, our observations are completely reversed in the case of glass transition scenario in our mixture because of the depletion interaction due to the presence of the external potential. We do not find a localization transition since the density of our system is much lower compared to the density of the system in which Voigtmann and Horbach27 observed the localization transition. The effect of density on these results will be investigated in the future. In the present study, different types of particles in our binary mixture have same mass, though their sizes are different. The masses of the two components are kept same to separate out the effect of depletion interaction on their dynamics. However, we do not expect any changes in the qualitative behavior of their dynamics, if the masses are different as in real systems, though there will be modifications in the exact val-
In conclusion, we have studied the dynamics of binary mixtures subjected to an external potential barrier. We have observed that the existence of depletion interaction due to the presence of repulsive potential makes pronounced difference in the long time dynamics of individual species compared to the binary mixture without the influence of any external potential. The depletion interaction causes the smaller particles to undergo a glass transition as we decrease the temperature even though the density is low. The larger particles undergo normal diffusion even at very low temperatures due to depletion interaction between the larger particles and the potential barrier. This result is interesting, as we can increase the transport of particles over a barrier by adding smaller particles in to the system, which will have consequences in many phenomena which involve crossing a barrier such as activated diffusion, chemical reactions, etc. Our results suggest that depletion interaction will accelerate these activated processes. It will be interesting to investigate Kramer’s barrier crossing problem28 in the presence of depletion interactions in the light of present results. Also these results contribute to our understanding of the kinetic selectivity of a particular species from a mixture by porous materials such as zeolites,29, 30 metal oxide frame works,31, 32 and biological channels.33, 34 It will also be interesting to investigate more complex potential energy landscapes and the depletion interactions caused by them and subsequent changes in the dynamical properties of the colloids. 1 W.
B. Russel, D. A. Saville, and W. R. Schowalter, Colloidal Dispersions (Cambridge University Press, USA, 1999). 2 L. J. Kaufman and D. A. Weitz, J. Chem. Phys. 125, 074716 (2006). 3 A. M. Puertas, E. Zaccarelli, and F. Sciortino, J. Phys.: Condens. Matter 17, L271 (2005). 4 F. Sciortino, S. Mossa, E. Zaccarelli, and P. Tartaglia, Phys. Rev. Lett. 93, 055701 (2004). 5 R. L. Hoffman, J. Rheol. 42, 111 (1998). 6 H. M. Laun, R. Bung, S. Hess, W. Loose, O. Hess, K. Hahn, E. Hadicke, R. Hingmann, F. Schmidt, and P. Lindner, J. Rheol. 36, 743 (1992). 7 H. N. W. Lekkerkerker, Colloids and the Depletion Interaction (Springer, New York, 2011). 8 F. Oosawa and S. Asakura, J. Chem. Phys. 22, 1255 (1954). 9 A. Vrij, Pure Appl. Chem. 48, 471 (1976). 10 K. Park, H. Koerner, and R. A. Vaia, Nano Lett. 10, 1433 (2010). 11 T. G. Mason, Phys. Rev. E 66, 060402 (2002). 12 R. Piazza, Curr. Opin. Colloid Interface Sci. 5, 38 (2000). 13 T. Odijk, Biophys. Chem. 73, 23 (1998). 14 Z. Dogic, K. R. Purdy, E. Grelet, M. Adams, and S. Fraden, Phys. Rev. E 69, 051702 (2004). 15 T. H. Lin and G. J. D. Phillies, J. Phys. Chem. 86, 4073 (1982). 16 G. S. Ullmann, K. Ullmann, R. M. Lindner, and G. D. J. Phillies, J. Phys. Chem. 89, 692 (1985). 17 R. L. C. Vink, A. De Virgiliis, J. Horbach, and K. Binder, Phys. Rev. E 74, 031601 (2006). 18 R. Roth and S. Dietrich, Phys. Rev. E 62, 6926 (2000). 19 A. D. Dinsmore, P. B. Warren, W. C. K. Poon, and A. G. Yodh, Europhys. Lett. 40, 337 (1997). 20 R. Roth, M. Rauscher, and A. J. Archer, Phys. Rev. E 80, 021409 (2009).
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J. Turro, X. Lei, C. C. Cheng, D. R. Korbin, and L. Abrams, J. Am. Chem. Soc. 107, 5824 (1985). 30 S. Sarkar, A. V. A. Kumar, and S. Yashonath, J. Chem. Phys. 112, 965 (2000). 31 J. Liu, J. Tian, P. K. thallapally, and B. P. McGrail, J. Phys. Chem. C 116, 9575 (2012). 32 R. Banarjee, A. Phan, B. Wang, C. Knobler, H. Furukawa, M. O’Keeffe, and O. M. Yaghi, Science 319, 939 (2008). 33 D. Boda, D. D. Busath, B. Eisenberg, D. Henderson, and F. Nonner, Phys. Chem. Chem. Phys. 4, 5154 (2002). 34 M. L. Lopez, M. Aguilella-Arzo, V. M. Aguilella, and A. Alcaraz, J. Phys. Chem. B 113, 8745 (2009).
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THE JOURNAL OF CHEMICAL PHYSICS 141, 034904 (2014)
Anomalous dynamics of binary colloidal mixtures over a potential barrier: Effect of depletion interaction A. V. Anil Kumara) School of Physical Sciences, National Institute of Science Education and Research, IOP Campus, Bhubaneswar 751005, India
(Received 14 May 2014; accepted 3 July 2014; published online 21 July 2014) The dynamics of a binary colloidal mixture under the influence of an external potential barrier has been studied by molecular dynamics simulations. The attractive depletion interaction between the barrier and larger particles fastens the dynamics of the larger particles over the potential barrier. At low temperatures, depletion interactions cause the larger particles to diffuse faster than smaller particles, which is counterintuitive. The repulsive barrier leads the small particles to undergo an anomalous diffusion which resembles the dynamics of systems undergoing a glass transition, while the larger particles undergo normal diffusion even at very low temperature. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4890282] I. INTRODUCTION
Colloidal dispersions consist of mesoscopic particles dispersed in a medium of microscopic particles and are typically characterized by the dynamics of the dispersed particles.1 There are easy ways of tuning the interaction potential between the colloidal particles, which make them act as model systems to study the fundamental questions on the properties of condensed matter systems such as equilibrium and nonequilibrium phase transitions. The colloidal systems themselves exhibit a number of interesting physical phenomena such as repulsive and attractive glasses,2, 3 cluster fluids,4 shear thinning and thickening transitions,5, 6 etc. Many of these phenomena are caused by the indirect interactions between the colloidal particles. Depletion interaction is the prominent indirect interaction, which arises when particles of different sizes are present in a colloidal dispersion.7 In a suspension of large and small particles, there will be a restricted volume region surrounding the big particles where small particles cannot enter. Overlap of these restricted volumes increases the volume accessible to small particles, which causes an effective attraction between the larger particles. The range and depth of this attractive interaction depends on the size and density of the small particles. This purely entropic interaction was first proposed by Asakura and Oosawa8 for a colloid-polymer mixture and later elaborated by Vrij.9 These interactions play a crucial role in the current understanding of the phase behavior and dynamics of dispersions containing colloids of disparate sizes and has been called upon to account for many phenomena such as size and shape selectivity,10, 11 protein crystallization,12 macromolecular crowding,13, 14 and dynamics in crowded medium.15, 16 In a binary colloidal mixture, subjected to an external potential, depletion interactions not only arise due to the overlap of excluded volumes of larger particles, but also due to the overlap of excluded volumes of larger particles and that of a repulsive potential. It has been shown that when a cola) Electronic mail: [email protected]
0021-9606/2014/141(3)/034904/6/$30.00
loidal binary mixture comprising particles of disparate sizes has been confined between the hard walls, the larger particles get attracted towards the wall resulting in an increased density of them near the wall.17–19 This can be explained on the basis of depletion interactions, according to which the hard wall also possesses a depletion layer surrounding it, in which the smaller particles cannot enter. Therefore, the larger particles get attracted towards the wall due to the overlap of the excluded volumes of the wall and the larger particles. This phenomena is also true for walls with a softer, but infinite confining potential. These observations led to the suggestion that whenever a binary mixture is subjected to an external repulsive potential, there will be an effective attraction for larger particles towards the potential due to depletion interactions. This conjecture has been verified by dynamical density functional theory calculations (DDFT)20 as well as Monte Carlo simulations.21 These investigations have revealed that because of the attractive depletion interactions, the density of large particles increases in the region of external repulsive potential and the strength of this depletion interaction depends on various parameters like density and number ratio of the mixture, size ratio of the particles, and width and height of the potential barrier. However, these studies were restricted to the equilibrium properties of the binary mixture. In this article, we investigate the dynamical properties of a colloidal binary mixture subjected to an external repulsive barrier, using molecular dynamics simulations. Our results reveal that the dynamical properties of this mixture are as rich and significant as the equilibrium properties. The depletion interactions fasten the dynamics of larger particles in the binary mixture and make them diffuse faster than the smaller particles as the temperature is lowered which is counterintuitive. The smaller particles undergo a typical glass transition scenario at lower temperatures even though the density is quite low. II. THE MODEL AND SIMULATION DETAILS
We performed canonical ensemble molecular dynamics simulations of a binary mixture of purely repulsive soft 141, 034904-1
© 2014 AIP Publishing LLC
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spheres, with interaction potential Vij (rij ) given by
III. RESULTS AND DISCUSSION
To investigate the effect of depletion interactions due to the presence of external potential barrier on the dynamics of binary mixture, we have carried out a number of molecular dynamics simulations at a fixed total volume fraction φ = 0.20, varying the volume fraction of the individual components. We started with a single component fluid of larger particles and in the subsequent simulations increased the volume fraction of smaller particles keeping the total volume fraction constant. The self-diffusion coefficients were calculated from the mean squared displacement of the particles. Figure 1(a) shows the self-diffusion coefficients of larger and smaller particles against the volume fraction of smaller particles, φ s . Dzl (Dzs ) is the self-diffusion coefficient of the larger (smaller) particles along the z-direction where the particles have to encounter the potential barrier, while D0l (D0s ) is the average self-diffusion coefficients along x and y directions where there is no external potential. For the smaller particles, the self-diffusion coefficient across the barrier is always lower than the self-diffusion coefficients along other two directions, which is expected since the particles have to cross the potential barrier along z-direction. However, for the larger particles the scenario is completely different. In the case of single component fluid of larger particles, again Dzl is lesser than D0l . As
s
D0
s
Dz
D
where i, j = l, s. We chose σ ss = 1.0, σ ll = 2.0, ss = 1.0, and ll = 4.0.21 All the parameters are expressed in dimensionless reduced units. Cross interaction parameters are obtained by Lorentz-Berthelot additive mixture rules. We kept all the masses equal, ms = ml = 1.0. The mixture is subjected to a potential barrier at the center of the simulation box along the z-axis, which is of Gaussian form, z − z 2 0 . (2) Vext (z) = εext exp − w Here, w is the range of external potential and εext is the height of the potential. We used w = 3.0 and εext = 2.0. Recent advances in experimental techniques make it possible to create such localized potential barriers.22 The simulations solve the system of first order differential equations q˙ i = pi /m, p˙i = Fi − λpi , where λ is the thermostat factor determined by the Gaussian principle of least constraint,23 qi and pi are the position and momentum vectors, respectively, and Fi is the net force acting on the particle i. The above equations are solved simultaneously using a fifth order Gear predictorcorrector method.24 We used a cubic simulation box of length Lb = 20.48 and periodic boundary conditions are imposed along all the three directions. We used a time step dt = 0.001 in reduced units (t0 = [ms σss2 /ss ]1/2 ). In each simulation run, we allowed the system to equilibrate for 5× 106 time steps and then the measured properties are averaged over a production run of 20× 106 steps. The dynamical properties are calculated from the trajectories produced in each production run. These properties are averaged over four simulation runs starting from different initial configurations.
l
Dz
1
(1)
0.5
0
0
0.05
φs
0.1
large small
(b)
1 Dz/D0
Vij (rij ) = εij (σij /rij )36 ,
l
D0
(a)
0.8 0.6 0.4 0.2
0
0.05
φs
0.1
FIG. 1. (a) Self-diffusion coefficients of small and large particles over the potential barrier and parallel to the potential barrier against the volume fraction, φ s of small particles. The larger particle volume fraction is φ − φ s . Here, φ is fixed at 0.20. Dzl and Dzs are the self-diffusivities of large and small particles along the z-direction. D0l and D0s are the self-diffusivities in the x − y plane. (b) The ratio of self-diffusion coefficients over the barrier to that parallel to the barrier for both small and large particles.
we increase the volume fraction of the smaller particles, the difference between D0l and Dzl starts decreasing and they become almost similar when we have a significant volume fraction smaller particles present in the mixture. The introduction of smaller particles into the system invokes the depletion interaction among the larger particles as well as among the large particles and the potential barrier. So the large particles feel an attraction towards the barrier which will enable them to cross the barrier easily. This increases Dzl and the larger particles diffuse as if they do not feel the presence of the barrier at all. This can be easily seen from Figure 1(b), where we have plotted the ratio Dzs /D0s and Dzl /D0l against φ s . Dzs /D0s remains much lower than one for all values of φ s which is as expected. However, Dzl /D0l starts at value of 0.75 and increases as φ s increases and becomes close to one at φ s = 0.10. At even higher volume fractions of smaller particles, i.e., above φ s = 0.10, the ratio starts decreasing. Earlier Monte Carlo simulations have shown that the depletion interaction for this system increases as φ s increases, goes through a maximum at φ s = 0.10 and then decreases.21 The results of molecular dynamics simulations agree with these observations. We concentrated on an equi-volume mixture with total volume fraction, φ = 0.20 for further investigations, since the depletion interaction is a maximum for this mixture. To investigate the effect of the potential barrier on the dynamics of the particles, we calculated the self-diffusion-coefficients along the potential barrier and perpendicular to it. Figure 2 depicts the temperature variation of the self-diffusion coefficients, Dzl , D0l , Dzs , and D0s . D0l and D0s show the typical temperature dependence for fluids. The smaller particles diffuse faster than the larger particles at all temperatures studied. However, the diffusion along the z-direction where the
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2.5
1
(a)
T=2.0 T=1.0 T=0.7 T=0.5 T=0.4 T=0.3 T=0.25
2 ρl(z)
0.1 z
D
Dl
Ds
0.01
1.5 1
z
D0 D0
0.5
l s
0
0.001 0
0.5
1
1.5
2
0
5
10 z
15
20
10 z
15
20
2.0
2.5
T
(b)
FIG. 2. Self-diffusivities of small and large particles along z-axis and in the x − y plane against temperature. The diffusivities along the z-direction show a crossover at T = 0.7.
ρs(z)
1.5 1.0 0.5 0 0
5
FIG. 3. Density profile as a function of z for (a) large particles and (b) small particles.
and y are uniform and do not show any deviations from the bulk density. We now turn our attention to the detailed dynamics of particles along the z-direction where they have to overcome the external potential barrier. Figure 4 displays the mean-squared displacements of both the species along the z-direction at various temperatures. At higher temperatures, smaller particles diffuse normally over the whole time and the mean-squared displacement increases linearly with time. However, at low temperatures we observe a standard glasstransition scenario: after the initial ballistic regime, there is a plateau which indicates localization of particles at the intermediate time, before diffusive region is observed at long 4
10
(a)
2
δrl
2
10
T=2.0 T=1.0 T=0.7 T=0.5 T=0.4 T=0.3 T=0.25
0
10
-2
10
-1
10
0
10
1
2
10
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3
10
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10
t 4
10
(b)
2
2
10 δrs
particles have to overcome the external potential barrier shows interesting temperature dependence. At high temperatures, the smaller particles diffuse faster than the bigger particles, which is expected. However, as the temperature is lowered, Dzs decreases more strongly compared to Dzl . As the temperature is lowered the smaller particles find it increasingly difficult to surmount the potential barrier and their diffusion coefficient decreases drastically. However, the larger particles feel the attraction towards the barrier due to depletion interaction and this attraction fastens their crossing over the barrier. Hence, they do not slow down appreciably and still retain the diffusive regime. Among the remarkable features of these results is that the extent of drop in the self-diffusivity of smaller particles relative to that of larger particles leads to the larger particles diffusing faster compared to smaller particles below T = 0.7. As we go down further, Dzs decreases rapidly, indicating localization of particles. The above observations suggest that the depletion interaction causes the larger and smaller particles to experience the potential barrier differently. This can be visualized by calculating the density profile of large and small particles along zdirection, ρ(z). Figure 3 shows the density profile of large and small particles at different temperatures. It is evident from the figure that the larger particles get attracted towards the barrier and smaller particles get depleted from it. As temperature decreases, the density of larger particles increases near the barrier and slowly the peak at the barrier splits in to two. The density profile of smaller particles exhibits a contrasting behavior. Within the region of the potential barrier, their density is much smaller than the bulk density and this difference increases as we decrease the temperature. The potential of mean force is given by −kB Tlnρ(z), the derivative of which with respect to z gives the effective force the barrier exerts on the particles. The figure shows clearly that the effective force on the smaller particles is increasingly repulsive as the temperature goes down. For the larger component, the force is attractive. At lower temperatures, when the density profile peak splits in to two, the potential of mean force has a small barrier at the middle, which results in a small repulsive effective force. This explains the slight reduction in the diffusivity of larger particles at low temperatures. We should also mention that the density profiles for both large and small particles along the x
0
10
-2
10
-1
10
0
10
1
2
10
10
3
10
4
10
t FIG. 4. Mean squared displacement of (a) large and (b) small particles along the z-direction.
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J. Chem. Phys. 141, 034904 (2014)
4
0.5
10
(a)
(a)
0.4
2
T=2.0 T=1.0 T=0.7 T=0.5 T=0.4 T=0.3 T=0.25
0
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T=2.0 T=1.0 T=0.7 T=0.5 T=0.4 T=0.3 T=0.25
0.3
l
δrl,0
2
αz (t)
10
0.2 0.1 0 -1 10
4
10
0
10
1
10
3
10
4
10
t
t 1.5
4
10
(b)
(b)
1.25
2
1.0 s
2
αz (t)
10 δrs,0
2
10
0
10
0.75 0.5 0.25
-2
10
-1
10
0
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2
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3
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0 -1 10
4
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t
t FIG. 5. Mean squared displacement of (a) large and (b) small particles averaged along the x- and y-directions.
FIG. 6. Non-Gaussian parameter α z as a function of time for (a) large particles and (b) small particles.
times. The time window over which this plateau extends increases as the temperature decreases. For larger particles, the mean-squared displacement is linear with time in the whole temperature regime we have investigated. This should be contrasted with the average mean squared displacement along x- and y-directions, where the particles do not need to encounter any potential barrier. This is depicted in Figure 5. Mean squared displacement of both large and small particles are linear in time and does not show any signatures of slowing down. This behavior makes it clear that the potential barrier is responsible for the dynamical slowing down of smaller particles. This should have been the case for larger particles also, however, the attractive depletion interaction between the larger particle and potential barrier keeps the particle in the diffusive regime. The deviation between the measured mean squared displacement and what is expected from normal diffusive behavior can be ascertained by the non-Gaussian parameter α z (t), defined by25
termediate times it increases go through a maximum and at large times it becomes again close to zero. The intensity of the maximum in α z (t) increases with decrease in temperature. Also the peak in α z (t) moves towards higher values of t as we decrease the temperature. This behavior of α z (t) is typical of super cooled liquids near to glass transition and completely in agreement with the conclusions made based on the behavior of mean squared displacements. The above observations are further confirmed by the time evolution of z-component of intermediate scattering function Fs (k, t) . We calculated the intermediate scattering function for the motion along the z-direction as N 1 exp[ik(zj (t) − zj (0))] (4) Fs (k, t) = N j =1
αz (t) =
1 z4 − 1, 3 z2 2
(3)
where z4 and z2 are the fourth and second moment of particle displacements along the z-direction, respectively. We have calculated α z (t) for both large and small particles at various temperatures. For large particles, α z (t) is very close to zero in the entire time range we calculated, which means the particle dynamics is diffusive all the time. This is true for all the temperatures at which α z (t) is calculated. However, in the case of smaller particles we observe two scenarios with respect to the temperature (Fig. 6). At high temperature, again the non-Gaussian parameter is close to zero in the entire time range and the particle dynamics is diffusive. As we decrease the temperature to 0.7 and below, there are three distinctive regimes in α z (t). At short times, α z (t) is close to zero, at in-
with k = 2π /Lb . For larger particles, the intermediate scattering function Fs (k, t) exhibits an exponential one-step decay, which is expected for liquids, for all the temperatures studied (Fig. 7). However for smaller particles, again we observe two scenarios with respect to temperature. At high temperature, the time evolution of Fs (k, t) is a one-step exponential decay. However, as we decrease the temperature to t = 0.7 or below, a plateau develops at intermediate times, followed by a non-exponential decay to zero. Such behavior is typical of glass forming liquid. These observations are in complete agreement with the behavior of mean squared displacement and non-Gaussian parameter. There are few comments in order here. First, our results for the dynamics of smaller particles are consistent with the experimental and molecular simulation results of DalleFerrier et al.26 They have studied the dynamics of dilute colloidal suspensions in a sinusoidal potential. It has been shown that the external potential leads the particle dynamics to undergo a glass transition scenario even though the density is very small. In our simulations, we observe similar behavior
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J. Chem. Phys. 141, 034904 (2014)
ues of self-diffusion coefficients. Our results could be experimentally verified as the recent experimental advances make it possible to realise microscopically localized potential barriers using optical tweezers.22
1.0 (a) T=2.0 T=1.0 T=0.7 T=0.5 T=0.4 T=0.3 T=0.25
0.6
l
Fs (k,t)
0.8
0.4
IV. CONCLUSIONS
0.2 0 -1 10
0
10
1
2
10
10
3
4
10
10
t 1.0 (b)
0.6
s
Fs (k,t)
0.8
0.4 0.2 0 -1 10
0
10
1
2
10
10
3
10
4
10
t FIG. 7. Intermediate scattering function Fs (k, t) for the motion along the zdirection as a function of time for (a) large particles and (b) small particles. The value of k used is 2π /Lz .
for the small particle dynamics, while the large particles do not show any characteristics of dynamics of system approaching glass transition. This supports our earlier argument that the larger particle, because of the attraction due to depletion interaction do not feel the presence of the potential barrier while smaller particles feel a heightened barrier due to repulsion. Second, the comparison of our results with earlier simulation results of Voigtmann and Horbach,27 in which they considered a binary mixture of disparate sizes, without any external potential reveals an interesting point. Their simulation indicated a double transition scenario, where the larger particle undergoes a typical glass transition dynamics first, as the density is increased. Small particle dynamics will be still diffusive and undergoes a localization transition if the density is increased further. Here in our simulations, we used temperature as the control parameter instead of particle density. As we decrease the temperature, it is the smaller particle which undergoes the slowing down rather than the bigger particles. Even at the lowest temperature we simulated, i.e., T = 0.25, we have observed that the larger particles undergo normal diffusive dynamics. Thus, our observations are completely reversed in the case of glass transition scenario in our mixture because of the depletion interaction due to the presence of the external potential. We do not find a localization transition since the density of our system is much lower compared to the density of the system in which Voigtmann and Horbach27 observed the localization transition. The effect of density on these results will be investigated in the future. In the present study, different types of particles in our binary mixture have same mass, though their sizes are different. The masses of the two components are kept same to separate out the effect of depletion interaction on their dynamics. However, we do not expect any changes in the qualitative behavior of their dynamics, if the masses are different as in real systems, though there will be modifications in the exact val-
In conclusion, we have studied the dynamics of binary mixtures subjected to an external potential barrier. We have observed that the existence of depletion interaction due to the presence of repulsive potential makes pronounced difference in the long time dynamics of individual species compared to the binary mixture without the influence of any external potential. The depletion interaction causes the smaller particles to undergo a glass transition as we decrease the temperature even though the density is low. The larger particles undergo normal diffusion even at very low temperatures due to depletion interaction between the larger particles and the potential barrier. This result is interesting, as we can increase the transport of particles over a barrier by adding smaller particles in to the system, which will have consequences in many phenomena which involve crossing a barrier such as activated diffusion, chemical reactions, etc. Our results suggest that depletion interaction will accelerate these activated processes. It will be interesting to investigate Kramer’s barrier crossing problem28 in the presence of depletion interactions in the light of present results. Also these results contribute to our understanding of the kinetic selectivity of a particular species from a mixture by porous materials such as zeolites,29, 30 metal oxide frame works,31, 32 and biological channels.33, 34 It will also be interesting to investigate more complex potential energy landscapes and the depletion interactions caused by them and subsequent changes in the dynamical properties of the colloids. 1 W.
B. Russel, D. A. Saville, and W. R. Schowalter, Colloidal Dispersions (Cambridge University Press, USA, 1999). 2 L. J. Kaufman and D. A. Weitz, J. Chem. Phys. 125, 074716 (2006). 3 A. M. Puertas, E. Zaccarelli, and F. Sciortino, J. Phys.: Condens. Matter 17, L271 (2005). 4 F. Sciortino, S. Mossa, E. Zaccarelli, and P. Tartaglia, Phys. Rev. Lett. 93, 055701 (2004). 5 R. L. Hoffman, J. Rheol. 42, 111 (1998). 6 H. M. Laun, R. Bung, S. Hess, W. Loose, O. Hess, K. Hahn, E. Hadicke, R. Hingmann, F. Schmidt, and P. Lindner, J. Rheol. 36, 743 (1992). 7 H. N. W. Lekkerkerker, Colloids and the Depletion Interaction (Springer, New York, 2011). 8 F. Oosawa and S. Asakura, J. Chem. Phys. 22, 1255 (1954). 9 A. Vrij, Pure Appl. Chem. 48, 471 (1976). 10 K. Park, H. Koerner, and R. A. Vaia, Nano Lett. 10, 1433 (2010). 11 T. G. Mason, Phys. Rev. E 66, 060402 (2002). 12 R. Piazza, Curr. Opin. Colloid Interface Sci. 5, 38 (2000). 13 T. Odijk, Biophys. Chem. 73, 23 (1998). 14 Z. Dogic, K. R. Purdy, E. Grelet, M. Adams, and S. Fraden, Phys. Rev. E 69, 051702 (2004). 15 T. H. Lin and G. J. D. Phillies, J. Phys. Chem. 86, 4073 (1982). 16 G. S. Ullmann, K. Ullmann, R. M. Lindner, and G. D. J. Phillies, J. Phys. Chem. 89, 692 (1985). 17 R. L. C. Vink, A. De Virgiliis, J. Horbach, and K. Binder, Phys. Rev. E 74, 031601 (2006). 18 R. Roth and S. Dietrich, Phys. Rev. E 62, 6926 (2000). 19 A. D. Dinsmore, P. B. Warren, W. C. K. Poon, and A. G. Yodh, Europhys. Lett. 40, 337 (1997). 20 R. Roth, M. Rauscher, and A. J. Archer, Phys. Rev. E 80, 021409 (2009).
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