https://doi.org/10.1007/s12039-019-1606-9 REGULAR ARTICLE
Impact of length scale of attraction on the dynamical heterogeneity: a molecular dynamics simulation study
SHASHANK PANTa,b and PRADIP KUMAR GHORAIa,∗
aDepartment of Chemical Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur, West Bengal 741 256, India
bBeckman Institute of Advanced Science and Technology, University of Illinois, Urbana-Champaign, IL 61801, USA
E-mail: pradip.ghorai@gmail.com
MS received 12 November 2018; revised 30 January 2019; accepted 11 February 2019; published online 25 March 2019 Abstract. We investigate the role of the length scale of attraction on the dynamical heterogeneity in three- dimensional core-softened (CS) model liquid having interaction potential of two different length scales. Using the simple CS model, we have shown that the presence of heterogeneity is attributed to the existence of attractive interactions in a system. The time scale associated with the peak position of the non-Gaussian parameter is found to be strongly dependent on the temperature for different length scale of attraction. The dynamical heterogeneity dramatically decreases with the increase in temperature. We further characterize dynamical heterogeneity by calculating new non-Gaussian parameter which indicates the existence of much longer relaxation timescale than that suggested by the peak position in the non-Gaussian parameter. Furthermore, we have shown the presence of particles, whose displacement distribution deviates from the Gaussian distribution and contributes towards the origin/presence of heterogeneity.
Keywords. Core-softened liquid; molecular dynamics simulation; heterogeneity; non-Gaussian parameter.
1. Introduction
Dynamical heterogeneity is a phenomenon seen in many liquids under different conditions. When a system con- sists of mobile and immobile particles, the relaxation rates of the particles are very different in different domain.1–9 Several studies show the dependence of dynamical heterogeneity on temperature by calculating alpha parameter, intermediate scattering function, van Hove correlation function, radial distribution function, etc., of mobile and immobile particles.10–17 In this arti- cle, we report the effect of the length scale of attraction on the dynamical heterogeneity. We use core-softened (CS) potential for three-dimensional model liquid. CS potential is an interaction potential with two length scales which is modelled by conjugating a Lennard- Jones potential with a Gaussian peak. Several authors extensively use this potential to reproduce water like anomalies.18–24
*For correspondence
The CS potential is given by25
UC S(ri j)=4εcs
σcs
ri j
12
− σcs
ri j
6
+aεexp
−1 c2
r i j−r0 σcs
2
(1)
whereri j = [ri− rj]is the distance between the two fluid particlesiand j.σcs andεcs correspond to the size and energy parameters respectively.r0 denotes the location of the Gaussian peak.candaare the parameters which control the width and height of the Gaussian part. The length scale of interaction can be tuned by varying the value of c. We use three different values of c which cor- respond to a system with repulsive interaction (system A), a system with attractive interaction (system B) and a system with more attractive interaction (system C).
Figure1shows three CS potentials used in the present study. Though CS fluid have been previously used to reproduce different thermodynamic anomalies present in water, it is not used to investigate the role of length scale interaction on dynamical heterogeneity.
1
Figure 1. Core-softened potentials with a different length scale of attraction used in the present study.
Table 1. 1/c2values for system A, system B and system C, respectively.
Name 1/c2 Nature
System A 2.0 Repulsive
System B 12.0 Attractive
System C 20.0 More Attractive
2. Model and simulation methodology
The liquid is modelled as a collection of spherical core-softened particles with an effective diameter σcs. Ini- tially, we start with a simulation box of volume V having Nnumber of fluid particles interacting with each othervia core-softened potential. In this study, the values of a and ro are 5.0 and 1.13 units respectively.18 As we are inter- ested to investigate the role of attractive interaction length scale on heterogeneity, we have used three different values ofcshown in Table1. All the parameters and quantities are expressed in reduced units as r∗ = r/σcs, U∗ = U/εcs, T∗ = kBT/εcs, P∗ = Pσcs3/εcs, ρ∗ = ρσcs3, time step dt∗ = dt(εcs/mσcs2)1/2. We perform all our simulations in the microcanonical ensemble with a cut off radius of 3.5σcs. The time step to integrate the equation of motion is 0.005. In order to fix the temperature of the systems, velocity scaling is used for the first 50,000 steps and further 50,000 steps are discarded. 5×106steps are used for production runs and all the data is stored after every 100 steps.
3. Results and Discussions
To detect the presence of dynamical heterogeneity, we compute the non-Gaussian parameterα2(t)defined as α2(t)= 3< δr4(t) >
5< δr2(t) >2 −1 (2)
where< δr4(t) >is defined as< (N1)N
j=1|rj(0,t)|4
>. δr(t) indicates the distance over which a particle moved in timet. rj(0,t) is the displacement vector for the jth particle. α2(t) reveals the deviation in the distribution of single particle displacement from a Gaus- sian distribution.16,26Figure2(a), (b) and (c) showα2(t) for system A, B and C respectively at different T∗. In system A, we do not observe any peak inα2(t)indicat- ing there is no dynamical heterogeneity associated with the system A. Further in system B and system C, we observe a distinct peak inα2(t)at a different time. This observation indicates the existence of dynamical hetero- geneity in the systems. For both system B and system C, the time scale associated with the peak position in α2(t) decreases with increase in temperature. This can be explained by the weakening of attractive forces with an increase in temperature for both system B and sys- tem C.
It is commonly accepted that α2(t) derives contri- butions primarily from those particles which travel more distance than the Gaussian distribution of particle displacements would predict. Similar behavior of non- Gaussian parameter has been reported by mode coupling theory which takes the hopping motion of the particles into consideration.28Flenner and Szamel reported a new non-Gaussian parameter [γ(t)] which weighs the immo- bile particles that have not moved as far as Gaussian distribution would predict.28γ(t)] is defined as
γ (t)= 1
3 < δr2(t) >< 1
δr2(t) >−1 (3) Figure3(a) and (b) showγ(t) at various temperatures for system B and system C, respectively. We observe that the peak forα2(t)occurs at a time shorter than those for γ(t) at all temperatures. This suggests the presence of another relaxation timescale which is much longer than that suggested by the peak time inα2(t).
It has previously been reported thatα2(t)is zero when the motion of the particles is ballistic and it increases upon entering β relaxation region; therefore on long time scales, it goes to zero inα relaxation.17 To deter- mine this strong increase of α2(t) in the β relaxation region, as seen in Figure2for system B and system C, we calculate self part of the van Hove correlation function [Gs(r,t)]and compare it with that obtained from Gaus- sian approximation,Ggs(r,t). Here we assumeGsg(r,t) is given by
=
3
2π <r2(t) >
3/2 e
−2<r3r22(t)>
(4) where < r2(t) > is defined as mean square dis- placement of the particles. Figure 4(a) and (b) show [Gs(r,t)−Ggs(r,t)]/Ggs(r,t)for system B and system
Figure 2. Non-Gaussian parameter for (a) system A, (b) system B and (c) system C respectively for different temperatures.
Figure 3. The new non-Gaussian parameter for the (a) system B and (b) system C.
Figure 4. [Gs(r,t)−Ggs(r,t)]/Ggs(r,t)versus r∗ for (a) system B and (b) system C respectively for t =t∗at different T∗.
C respectively for t = t∗, where t∗ is the point of maxima in α2(t) in Figure 2(a) and (b). We observe that for small distances (r∗ < 0.4), the relative differ- ence betweenGs(r,t)andGgs(r,t)is less than a factor of 2, whereas for higher distance (r∗ < 0.4), Ggs(r,t) underestimatesGs(r,t)and this behavior increases with decreasing temperature. This represents that at a low temperature there are a significant number of particles
that have traveled farther than would be expected from the Gaussian approximation for CS liquids with attract- ing interaction.
Further, we definer∗as the point from which the dif- ference starts to become positive and further increases, thus atr =r∗,Gs(r,t)=Ggs(r,t). This helps to define mobile and immobile particles in the system. Thus, we define “mobile particle(Am)” as A particles that have
Figure 5. Spatial correlations [g(r∗)] for system B and C at different temperatures.
Figure 6. Spatial co-relations gAm−Am/gA−Afor system (a) system A and (b) respectively at T∗=0.35 and 0.50.
moved a distance r(r > r∗) in time t∗. We compute the spatial correlation between mobile particles and compare it with the bulk shown in Figure 5. With the incorporation of high attractive interactions in the CS fluid, there is an increase in the correlated motions of the particles, indicated by the peak height of g(r∗). The results suggest that for a short distance of 1.0 unit the mobile particles (gAm−Am) are highly correlated com- pared to that of the bulk particles(gA−A). Here ‘bulk’
refers to all of the A particles (all CS particles). Similar
trend is observed for both system B and system C. On decreasingT∗, the correlation between the mobile par- ticles for both the systems significantly increases. The spatial correlation ratio gAm−Am/gA−A of the particles for system B and system C is shown in Figure6. The relative height of the first shell quickly increases and the relative increase is more in the presence of shorter length scale of attraction as shown in Figure6(b). We do not observe any significant change for longer length scale of attraction shown in Figure6(a) on changing the
temperature. This observation shows that for CS fluid, the dynamical heterogeneity significantly depends on the length scale of attraction. With the decrease in length scale of attraction (system C), the relative height of the correlation ratio gAm−Am/gA−Aincreases. This indicates that the attractive force with shorter length scale primar- ily contributes towards heterogeneity.
4. Conclusions
We have shown the existence of dynamic heterogeneity in three-dimensional CS model liquid having interac- tion potential of two different length scales. Using the simple CS model, we have shown that the presence of heterogeneity is attributed to the existence of attractive interactions in a system. The dynamical heterogene- ity strongly depends on the length scale of attraction.
The absence of maxima in the non-Gaussian parame- ter for repulsive CS fluids with a longer length scale of interaction indicates that heterogeneity occurs due to the presence of attractive interactions. We have shown a strong dependence of the peak position of the non- Gaussian parameter on temperature for both system B and system C with a different length scale of attraction.
This indicates that dynamic heterogeneity dramatically decreases with temperature. We further characterize the heterogeneity by calculating new non-Gaussian param- eter, which indicates the existence of much longer relaxation timescale than that suggested by the peak time inα2(t). Furthermore, by calculating the relative differ- ence ofGs(r,t)with respect toGsg(r,t), we have shown the presence of particles whose displacement deviates from the Gaussian distribution and contributes towards the origin/presence ofheterogeneity.
Acknowledgements
P. K. G. thanks CSIR (Project Number: 01(2558)/12/EMR-II) for financial support.
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