How Dimensionality Affects the Structural Anomaly in a Core-Softened Collo

1. Introduction

There are a large number of models for fluids and colloids, ranging from ab initio and atomistic molecular dynamics simulations to continuum models. In between this extrema, symmetric core-softened (CS) potentials are a class of effective models largely employed to study competitive systems, where the two characteristic length scales in the effective interaction between the particles try to dominate the fluid structure [1]. Usually, colloids are made of molecular subunits which form a central packed agglomeration and a less dense, more entropic peripheral area. With these, colloidal particles show distinct conformations that compete to rule the suspension behavior [2]. In this sense, they are usually modeled using isotropic CS potentials [3,4].

Although isotropic, the CS class of potentials shows a competition that come from the existence of two characteristic length scales in the potential [5,6,7,8,9,10,11,12,13,14] or from softened repulsive potentials [15,16,17,18,19]. As examples, experimental works have shown that the effective interaction in solutions of pure or grafted PEG colloids and Ficoll are well described by two length scales potentials [20,21,22,23,24]. Likewise, computational studies indicate the same type of effective interactions for polymer-grafted nanoparticles [25,26] or star polymers [27].

Water, the most common solvent for colloidal suspensions and the most important material in Earth, also shows a competition between two structures that originates from the hydrogen bonds breaking and forming between the water molecules [28,29,30]. As a consequence, water shows more than 70 anomalous behaviors due this struggle between two conformations trying to dominate the fluid structure [31]. Anomalies are properties that differ from the expected and observed in most materials. Certainly, the most well known water anomaly is the density anomaly: normal liquids contract upon cooling at constant pressure, while anomalous fluids expand as the temperature decreases. This anomaly leads to the maximum value in the water density, 1g/cm³ at 4^oC and constant pressure of 1 atm [32]. Also, anomalous fluids get disordered as the density increases – the opposite from the expected: a higher order in the system as the density grows [33]. In this sense, CS potentials are simple effective models that allow us to unveil the mechanisms behind the behavior of complex fluids as water and hard core - soft shell colloidal suspensions [34,35,36,37,38,39,40,41,42,43,44,45,46,47,48].

The shape and the type of interaction between colloidal particles combined with the presence of distinct confining surfaces have key roles in self-assembly, phase separation, interfacial activities and anomalous behavior of systems modeled by CS potentials [49,50,51,52,53]. Those peculiarities are in agreement with several experiments involving colloidal cluster, formation of self-assembled mesophases, layering packing and synthesis of new structures using amphiphilic molecules [54,55,56,57,58,59].

Additionally, structure of colloidal systems has been extensively studied by molecular dynamics and Monte Carlo simulations and the presence of patterns have been found for 3D bulk [42], 2D bulk [60,61,62] and confined systems [52,63,64,65,66,67,68,69,70,71]. Significant differences are reported when going from a 2D to a quasi-2D system. When confined in very small slit pores, CS potentials can present a 2D-like behavior, but growing the size of slit pores affects the solid-liquid transition at low densities due the out-of-plane motion of particles [71]. Likewise, the quasi-2D limit can affect the temperature of maximum density observed in CS models [72] and the crystalline structure [73].

Specifically, de Oliveira and co-authors proposed a CS ramp-like potential that presents thermodynamic, dynamic and structural anomalies obeying the same hierarchy observed in water [74,75]. Since then, effects of hydrophobic and hydrophilic confinement through plates and nanotubes on those anomalies and in solid-liquid and surface transitions also have been studied [76,77,78,79]. They show that entropic effect from flexibility of confining walls can break the dynamic and structural anomalies of the CS fluid [80], as well induce superdiffusive to diffusive first order phase transition [81]. On the other hand, rigid walls can induce surface phase transitions [82] and arise a new region of structural anomaly [83]. More recently, Bordin and Barbosa [84] studied the 2D case of this CS fluid. They found a second region of density, diffusion and structural anomaly at high densities. The first region of anomalies is associated with the competition between scales of CS potential, while the second one is generated by a reentrant fluid marking an order-disorder transition. Besides that, an inversion of hierarchy on these anomalies was observed, in agreement with results from Dudalov et. al [85]. Later, Cardoso and co-workers [62] found a third region of structural anomaly at higher densities. They show that each anomalous behavior can be associated to reentrant fluid phases that separates two solid crystalline regions, and to distinct aggregates size in the fluid phase.

But where is the boundary between the 3D behavior, with one region of structural anomaly in the fluid phase associated to a crystalline-amorphous solid transition [75,86] and the 2D behavior, where up to three anomalous structural regions was found [49,62]? To unveil this location, we expand our previous works [64,82,83] to see what happens in the quasi-2D limit with the CS model confined between two flat, highly repulsive solvophobic walls. Our findings indicate that the limit between 2D-like or 3D-like behavior is associated to the layering and with the existence of order-disorder transitions in the layers.

This paper is organized as follow: in section 2, we present the model; in section 3, we present the methods and simulation details; in section 4, we discuss the results; and, in section 5, we present the conclusions.

2. The Model

The colloidal system is modeled by spherical particles confined between smooth parallel plates. The particle-particle (P-P) interaction occurs through a two length scales potential, given by

$$\frac{U\left(r_{ij}\right)}{\epsilon }=4\left[{\left(\frac{\sigma }{r_{ij}}\right)}^{12}-{\left(\frac{\sigma }{r_{ij}}\right)}^6\right]+aexp\left[-\frac{1}{c^2}{\left(\frac{r_{ij}-r_0}{\sigma }\right)}^2\right]$$

(1)

where $r_{ij}=|{\overrightarrow{r}}_i-{\overrightarrow{r}}_j|$ is the distance between two particles, i and j. The first term is a standard Lennard-Jones (LJ) 12-6 potential where $\epsilon$ is the depth and $\sigma$ the particle’s diameter. The second one is a gaussian centered on radius $r_0$ and width c. We used the following parameters: $a=5$, $r_0/\sigma =0.7$ and $c=1$, that reproduces waterlike anomalies, like density, diffusion and structural anomalies [74,75]. The P-P interaction potential has a cutoff of $r_c/\sigma =3.5$.

The wall-particle (W-P) interaction is given by a strong repulsive potential, the so-called R6 potential [79,83]

$$U_{R6}\left(z\right)=\left\{\begin{array}{cc}4\epsilon {(\sigma /z)}^6+0.1875\epsilon (z/\sigma )-U_{R6c},\hfill & z\le z_c\hfill \\ 0,\hfill & z>z_c\hfill \end{array}\right.$$

(2)

where z is the distance between the particles and the walls, $z_c=2.0\sigma$ and $U_{R6c}=4\epsilon {(\sigma /z_c)}^6+0.1875\epsilon (z_c/\sigma )$. Figure 1 shows the profile of P-P and W-P interaction potentials. All quantities used in this work are given in LJ units [87], like for example, distance $r^{*}=r/\sigma$ and temperature $T^{*}=k_{B}T/\epsilon$. The symbol ^* will be omitted for simplicity.

3. Methods and Simulation Details

We perform molecular dynamics simulations in the $NVT$ ensemble. 2028 particles were confined between two parallel smooth fixed plates. The walls are held fixed in the z direction and separated by a distance $L_z$. $L_z$ was varied from $4.8$ to $6.2$. Those specific separations were chosen so we can study system with two or three particles layers. For each fixed $L_z$, we change the simulation box size, given by L, in order to simulate different total densities. Total densities are defined as $\rho =N/\left(L_{ze}{L}^2\right)$, where $L_{ze}=L_z-\sigma$ is the effective distance between the plates. Periodic boundary conditions were used in x and y directions. Our simulations were performed along the isotherm $T=0.150$ and we analyzed the effects of varying $\rho$ and $L_z$.

The equations of motion were integrated through Velocity Verlet algorithm, and the Nose-Hoover heat-bath with a coupling parameter Q = 2 was used to kept T fixed. Equilibration was reached after $2\times {10}^6$ time steps of simulation and followed by $1\times {10}^6$ time steps for the production of results. The time step was $\delta t=0.001$, and to ensure the equilibration, we track the lateral pressure and energy as function of time.

The system structure was analysed by the transversal density profile and the lateral radial distribution function, besides some correlated quantities, like translational order parameter, the two-body entropy and the cumulative two-body entropy. The lateral radial distribution function $g_{\Vert }\left(r_{\Vert }\right)$ is defined as

$$\begin{array}{c}\hfill g_{\Vert }\left(r_{\Vert }\right)\equiv \frac{1}{{\rho }^{2}V}\sum _{i\ne j}\delta (r_{\Vert }-r_{ij})\left[\theta \left(\left|z_i-z_j\right|\right)-\theta \left(\left|z_i-z_j\right|-\delta z\right)\right].\end{array}$$

(3)

where $r_{\Vert }$ is the parallel distance between particles in the x and y directions. Considering that systems structure themselves in layers, the Heaviside function, $\theta \left(x\right)$, restricts the sum of particle pairs in the same slab of thickness $\delta z/\sigma =1.0$ for contact and central layers.

The translational order parameter is defined as [88,89,90]

$$\tau =\int \mid g\left(ϵ\right)-1\mid dϵ,$$

(4)

where $ϵ=r_{\Vert }{\rho }_l^{1/2}$. We calculate the average number of particles per layer, $<N_l>$, and define the layer density as ${\rho }_l=<N_l>/\left(L^2\delta z\right)$.

The translational order parameter provides us information about the structure of the system. Gas and liquids have low values of $\tau$, while crystal and amorphous solids present high values of $\tau$. In normal fluids, $\tau$ increases monotonically with density or pressure at constant temperature, but in anomalous fluids $\tau$ decreases with density or pressure for some temperatures. The current potential presents one region of anomaly in the 3D bulk system [75], due the competition between scales. For confined systems and considering the particles near to the walls (contact layers), besides the anomalous region due the competition between scales, another one rises induced by nanoconfinement at low densities [83]. More recent works [62,84] showed that dimensionality leads to extra structural anomalies at higher densities and low temperatures.

The two-body entropy is a structural quantity. It is defined as [91]

$$s_2=-\frac{{\rho }_l}{2}\int \left[g_{\Vert }\left(r_{\Vert }\right)ln\left(g_{\Vert }\left(r_{\Vert }\right)\right)-g_{\Vert }\left(r_{\Vert }\right)+1\right]d^2{r}_{\Vert },$$

(5)

The normal behavior of $s_2$ is to decrease with density or pressure at constant temperature. However, in anomalous systems $s_2$ increases with the density or pressure at some region of its phase diagrams - the anomalous region. Like the translational order parameter, $s_2$ also presented a double region of anomaly for confined systems [83] and a triple anomalous region for the 2D case [62].

The last quantity analyzed was the cumulative two-body entropy, defined as [92]

$$C_{s2}\left(R\right)=-\pi {\int }_0^R\left[g_{\Vert }\left(r_{\Vert }\right)ln\left(g_{\Vert }\left(r_{\Vert }\right)\right)-g_{\Vert }\left(r_{\Vert }\right)+1\right]r_{\Vert }d{r}_{\Vert },$$

(6)

where R is the upper integration limit. This quantity give us information about the long range translational order. For fluids and amorphous phases, that do not present long range information, $C_{s2}\left(R\right)$ converges, while for solids, that present long range information, $C_{s2}\left(R\right)$ diverges.

4. Results

We can imagine the 2D limit as the single layer limit. In this sense, we have investigated systems with two, two-to-three or three layers of confined CS colloids, depicted by their transversal density profiles in Figure 2(a). Layers near to the walls are called contact layers, while the others, if exist, are called central layers. Systems with two well defined layers are observed for the most densities explored for plates separated by $L_z=4.8$ and $5.0$, while three well defined layers occur for $L_z=5.8$ to $6.2$. Intermediate values of $L_z$ lean over to present a transition between two-to-three layers for most densities simulated. In Figure 2(b) and (c) we show examples of snapshots of systems with two ($L_z=4.8$) and three ($L_z=6.0$) well defined layers, respectively.

Our first analysis is related to particles located near to the walls, the contact layers. As already mentioned, this model has one region of anomaly in translational order parameter, $\tau$, for 3D bulk system [75,86] due the competition between scales and three regions of anomaly for 2D bulk system [84], two of them due the competition between sales and another one due the order-disorder transition at intermediary ($\rho \approx 0.42$) and high ($\rho \approx 0.65$) densities [62]. Confining fixed walls also induce a new region of anomaly at low densities, ${\rho }_l<0.350$ [83]. Now, we present the behavior of $\tau$ for higher densities that studied in the past and we found the peculiar profile presented in Figure 3(a). It shows the parameter $\tau$ for the contact layer as function of layer density, ${\rho }_l$, for several separation of plates at $T=0.150$. Systems with plates separated by $L_z=5.2$, $5.3$, $5.5$ and $5.6$, drafted in orange lines, do not have anomaly for ${\rho }_l\approx 0.400$, that is, present a 3D-like fluid (or disordered) behavior. On the other hand, the another curves corresponding to $L_z=4.8$, $5.0$, $5.8$, $6.0$ and $6.2$, present a third region of anomaly, what means a 2D-like (ordered) behavior for this gap of densities ($0.400\le {\rho }_l\le 0.500$), including the same values of ${\rho }_l$ comparing to found for 2D systems [62,84].

Another quantity analyzed was the two-body entropy, that gives us structural information, because its definition is based on the radial distribution function (equation 5). Normal fluids lean over to present values of $s_2$ that monotonically decrease with density. $s_2$ also presents anomalous behavior in 3D and 2D bulk and 3D confined systems. Two regions of anomaly in $s_2$ were already shown in the past for ${\rho }_l$ until $\approx 0.350$. Now, going to higher layer densities, we found the profile seen in Figure 3(b). In current work we explore higher densities and we found a 2D-like and a 3D-like behavior, depending on the $L_z$ studied. Like observed for the translational order parameter, $s_2$ also presented a 2D-like behavior at high densities ($0.400\le {\rho }_l\le 0.500$) for $L_z=4.8$, $5.0$, $5.8$, $6.0$ and $6.2$ and a 3D-like behavior for $L_z=5.2$, $5.3$, $5.5$ and $5.6$.

To understand why some cases have one or more structural anomalous regions, and their origins, we analyze the transversal density profile ($\rho \left(z\right)\times z$), the lateral radial distribution function ($g\left(r_{\left|\right|}\right)\times r_{\left|\right|}$) and the cumulative two-body entropy ($|C_2\left(r_{\left|\right|}\right)|\times r_{\left|\right|}$) for three situations: the lowest separation $L_z=4.8$, corresponding to a system that present two layers, the separation $L_z=6.0$ that present three layers and a intermediate case, $L_z=5.5$, that has a two-to-three layers transition.

Let’s begin by analyzing the data for $L_z=4.8$, Figure 4. The solid green lines correspond to densities in the region of highly-ordered particles, $0.390\le {\rho }_l\le 0.441$. The dashed light green lines correspond to some immediately densities above the maximum in $\tau$ and minimum in $s_2$, while the dashed-dotted light green lines correspond to some immediately densities below the maximum in $\tau$ and minimum in $s_2$. In Figure 4(a), the transversal density profiles show two well defined layers, evidenced by zoom given in the inset graph, which suggests ordered particles. The Figure 4(b) shows the lateral radial distribution function and makes evident the ordering of particles in the third maximum of $\tau$ (solid green lines) comparing to vicinity densities (disorder-order-disorder transition), the same behavior observed in 2D system [62,84]. The long range translational behavior, given by the cumulative two-body entropy, $|C_2\left(r_{\left|\right|}\right)|$, confirms the tendency of solidification at densities $0.390\le {\rho }_l\le 0.441$.

The same behavior was found for another system that present 2D-like behavior at high densities, $L_z=6.0$, shown in Figure 5. The solid violet lines correspond to densities in the region of ordered particles (third maximum in $\tau$ and minimum in $s_2$), $0.381\le {\rho }_l\le 0.458$. The dashed magenta lines correspond to some densities immediately above the maximum in $\tau$ and minimum in $s_2$, while the dashed-dotted magenta lines correspond to some densities immediately below the maximum in $\tau$ and minimum in $s_2$. Although the fact that here the colloids are arranged in three well defined layers, like seen in Figure 5(a) and its inset graph, the basic mechanism behind the structural anomalous region is the same as the one found for the previous case, Figure 4. Both $g\left(r_{\left|\right|}\right)$ and $|C_2\left(r_{\left|\right|}\right)|$ indicate a order-disorder transition in the contact layers, as we can see in Figure 5(b) and (c), respectively. Therefore, the origin is the same as the one observed in the 2D system: a ordered-disordered transition leads to the waterlike structural anomaly.

So far, we have observed a 2D-like behavior and mechanism for the structural anomalies in systems with two and three layers. But what happens in systems with a single anomalous region? To clarify this point, we show in Figure 6 the $L_z=5.5$ case. The solid orange lines correspond to densities in the region that should present ordered particles (third maximum in $\tau$ and minimum in $s_2$ for another cases), $0.394\le {\rho }_l\le 0.465$. The dashed yellow lines correspond to densities slightly above, while the dashed-dotted yellow lines correspond to densities slightly below the region where it was supposed to happen the ordering of particles. In Figure 6(a), the transversal density profiles show a two-to-three layers transition. However, as indicated by the zoom in the inset graph, the density does not go to zero in the region between the layers. Unlike the previous cases (Figure 4 and Figure 5), here there is a connection between contact and central layers. This indicates that some particles from the central layer jump to the contact layer, and the other way around as well. This is a consequence of the frustration added by this specific separation length, that forces the colloids to organize themselves in such way that the separation between two layers is smaller than the second length scale in the CS potential but bigger than the first length scale. This frustration prevents the system to relax to an organized phase. It is clear also in Figure 6(b). The lateral radial distribution function does not go to zero for separations between the first and second peaks, preventing the ordered phase. This is corroborated by the cumulative two-body entropy, $|C_2\left(r_{\left|\right|}\right)|$, that presents a liquid (disordered) profile for all densities, like shown in Figure 6(c). Without order-disorder transition, the system leans to present a 3D-like behavior.

The central layer was observed for separations in the range $5.5\le L_z\le 6.2$. For these separations, the anomalous regions at high densities (around $\approx 0.4$) can be observed in the central layer for the cases the three layers are well defined - without particles hopping from one layer to another one.

5. Conclusions

Core-softened colloids can have unique and peculiar structural properties. The model employed in this study shows multiple regions of structural anomalous behavior in 2D, while the 3D case has only one. Our quest in this paper was to unveil the details about the structure of a CS model in the quasi-2D limit.

Our findings indicate the limit between 2D and 3D-like structural behavior for confined colloids: if the CS particles layers can accommodate themselves near one of the characteristic interaction length scales, the layers act as independent 2D systems – colloids from one layer do not jump to another one. On the other hand, if the confinement frustrates the system, does not allowing a inter-layer separation close to one of the potential length scale, the colloids move from one layer to another, no longer acting as independent layers, but approaching the 3D limit by having this translational hopping between the layers.