The Synchronous to Exchange Transition in Magnetically Driven Colloidal Dimers

Abstract

Numerical simulations are used to investigate the collective dynamics of an ensemble of driven paramagnetic colloidal particles confined between two plates. The particles are subjected to an external precessing field and, depending on the field frequency and cell thickness, they can assemble into rotating dimers, up and down binary crystals or dynamic states characterized by particle exchange between nearest dimers. We characterize the transition between the synchronous and the exchange state by measuring the fraction of “active” particles, namely colloids that do not stroboscopically recover their original place in a Voronoi cell. We find that the relaxation time decreases algebraically with the reduced frequency and that the transition displays signatures of an absorbing phase transition.

1 Introduction

Microscopic colloidal particles under time-dependent external fields represent an accessible model system to investigate the fascinating emergent dynamics that occur when many-body systems are driven out of equilibrium [1,2,3,4,5]. Colloids have a size in the visible wavelength, are characterized by experimentally accessible time scales, and can be easily manipulated with the aid of relatively low intensity external fields [6,7,8]. When the particles are located close to a wall, or within a narrow channel, the combination between pair interactions and confinement may lead to novel dynamics and emerging phenomena [9,10,11,12,13].

Here, we use numerical simulations to investigate the dynamic states emerging from a collection of magnetic particles strongly confined between two thin plates such that overpassing along the perpendicular direction is forbidden. A similar confinement was studied in the past to model geometric frustration, but it used size-tunable hydrogel particles without an external field, and thus the particles were passive and non driven [14,15,16]. Instead we consider the situation where, in the presence of a time-dependent field, the particles form a series of dynamic states resulting from the combination of excluded volume, confinement and induced magnetic dipolar forces. These states were recently realized experimentally [17, 18], and here we present numerical simulation results aiming at investigating the transition between two of them. In particular, we focus on the transition between a collection of synchronously rotating, localized dimers and an exchange phase, where the dimers break and exchange particles between them. We start the present contribution by illustrating the experimental system and the different dynamic states that the particles form by varying the field parameters. After that, we describe in detail the numerical simulation scheme adopted. Later we define our order parameter and how we extract different information on the nature of the observed transition. We finally conclude the manuscript by resuming the main results and discussing the nature of the observed transition.

2 Realization of the Colloidal Dimers and Exchange States

2.1 The Experimental System

The experimental system was developed in Ref. [17] and employed commercial paramagnetic colloids (Dynabeads M-270) made of a cross-linked polystyrene matrix with surface carboxylic groups. These particles have an average diameter $d=2.8 \,\mathrm {\mu m}$ and present a narrow size distribution. The magnetic properties of these particles arise from the uniform doping with iron oxide superparamagnetic grains ($\sim 20\%$ by vol.) which increases the particle density to $\rho = 1.6\, \mathrm {g \, cm^{-3}}$. Due to this doping, the particles can be controlled by an external magnetic field $\boldsymbol{B}$. In particular, when $\boldsymbol{B}\ne 0$, the paramagnetic colloids acquire an induced dipole moment which points along the field direction, $\boldsymbol{m} = \pi d^3 \chi \boldsymbol{B} /(6\mu _0)$, being $\chi$ the magnetic volume susceptibility of the particle, and $\mu _0$ the permeability of vacuum. Thus, a pair of particles (i, j) at a distance $r= |\boldsymbol{r_i}-\boldsymbol{r_j}|$ interacts via the magnetic dipolar potential,

$$\begin{aligned} U_{dip}=-\frac{\mu _0}{4 \pi r^5}\left[ 3 ( \boldsymbol{m}_i \cdot \boldsymbol{r})( \boldsymbol{m}_j \cdot \boldsymbol{r}) - ( \boldsymbol{m}_i \cdot \boldsymbol{m}_j) \boldsymbol{r}^2 \right] \end{aligned}$$

(7.1)

which is attractive (repulsive) for particles with magnetic moments parallel (perpendicular) to r.

The particles were diluted in highly deionized water, and confined between two glass surfaces made of a plain microscope slide and a coverslip. To achieve a small confinement, both plates were manually pressed, and later glued with a fast-curing epoxy adhesive. With this method, it was possible to obtain a small thickness in the range $h\in [3,6] \mathrm{\mu m}$. Such thickness was measured by analyzing the horizontal projection length of formed dimers under a static, perpendicular field $\boldsymbol{B}=B\boldsymbol{z}$. The particles were visualized with an upright optical microscope which was equipped with a set of custom build magnetic coils, that allows generating homogeneous, static and time-dependent magnetic fields.

Fig. 7.1

a–c Optical microscope images of confined paramagnetic colloids within a cell of thickness $h=3.9\, \mathrm {\mu m}$. In a there is no external field, in b the particles are subjected to a precessing field with amplitude $B_0=7.3$ mT, cone angle $\theta = 26. 9^{\circ }$, frequency $f = 1\,$Hz, and in c the frequency is raised to $f = 20\,$Hz. All images have a scale bar of $10 \,\mathrm {\mu m}$. d Left: Sketch of the particles inside a cell of thickness $d < h < 2d$ for frequency $f<f_r$ (top) and $f>f_r$ (bottom). Right: The precessing magnetic field with the cone angle $\theta$. e Normalized angular frequency $\nu$ of a dimer versus field frequency f showing the transition from synchronous to asynchronous regime at $f_c = 9.8$Hz. Scattered data are experimental results, the continuous line is from numerical simulation. Image adapted with permission from Ref. [17]

2.2 Colloidal States Under the Precessing Field

From Eq. 7.1 follows that the pair interactions between the paramagnetic colloids can be tuned by an external field $\boldsymbol{B}$. When the particles are confined above a plane, a static field $\boldsymbol{B}=B\boldsymbol{z}$ applied perpendicular to such plane induces an isotropic repulsion and, within the correct range of $\boldsymbol{B}$, one can induce the formation of a triangular lattice [19, 20]. In contrast, anisotropic attractive interactions can be induced via a static, in-plane field applied along the $\boldsymbol{x}$ or $\boldsymbol{y}$ axes [21, 22]. This situation becomes different when the particles are confined between two plates, as shown in Fig. 7.1d. Osterman et al. [23] demonstrated that the confinement may soften the pair repulsion. For particles enclosed between two hard walls and separated by a distance $h<2d$, the potential in Eq. 7.1 may be rewritten as,

$$\begin{aligned} U_{dip}=-\frac{\mu _0 m^2}{4 \pi }\left[ \frac{r^2-2z^2}{(r^2-z^2)^{5/2} }\right] \!\!, \end{aligned}$$

(7.2)

which shows that two particles repel when the elevation difference between their centers $\Delta z$, is $\Delta z<d/\sqrt{5}$, and otherwise they experience a short-range attractive and long-range repulsive potential. Such interactions give rise to different self assembled structures at equilibrium, featuring hexagonal, square, stripe or labyrinth-like ordering.

In contrast to a static field, the formation of interacting dimers was induced via a time-dependent field, as shown in Fig. 7.1d. This field performs a conical rotation around an axis $\boldsymbol{z}$ perpendicular to the sample plane with a frequency f and a cone angle $\theta$,

$$\begin{aligned} \boldsymbol{B} = B_0 [\cos {\theta } \boldsymbol{\hat{z}}+\sin {\theta }(\cos {(2 \pi f t)}\boldsymbol{\hat{x}}+ \sin {(2\pi f t)}\boldsymbol{\hat{y}}) ], \end{aligned}$$

(7.3)

where $B_0$ is the field amplitude. Under this type of forcing, novel dynamic colloidal patterns were observed. In particular, the sequence of images in Fig. 7.1a, c shows how a colloidal suspension confined to a narrow cell of thickness $h = 3.9 \,\mathrm {\mu m}$ self-organizes from an initial disordered phase (a) with $\boldsymbol{B}=0$. A precessing field with a relative slow frequency ($f=1$Hz, $\theta = 26.9^{\circ }$) arranges the paramagnetic colloids into an ensemble of rotating dimers, which perform a rotation around the z axis (b). Each dimer is composed of two particles, one closer to the top plate (“up”) and the other closer to the bottom one (“down”). They can be experimentally distinguished by their different brightness resulting from the different elevations, and are highlighted in the images by two colors. The dimers are stable as long as the field is kept fixed, and this state can be destabilized by increasing f. As shown in Fig. 7.1c, by raising the driving frequency to $f=20$Hz, the dimers break and the colloidal systems transforms into two separated lattices made of up and down particles. The up particles are close to the top plane, and remain there as long as the field is applied. A corresponding lateral sketch of the the particle locations with respect to the plates can be found in Fig. 7.1d. Depending on the density of the particles and the cell thickness, a continuous variation of f can drive the system into two different high frequency states. For each transition path, a transition frequency can be defined, namely $f_r$, which separates the stable dimer from the broken (up and down) state and a synchronous to asynchronous transition frequency, $f_c$. The latter separates two different rotational modes of the dimers, Fig. 7.1e. When $f<f_c$, the dimers rotate synchronously with the precessing field, and their rotational frequency is $\nu = f$. In contrast, for $f>f_c$, the phase-lag angle between $\boldsymbol{B}$ and the dimer long axis changes in time, and the dimers enter into an asynchronous regime, showing a characteristic “back-and-forth” rotation. Due to these oscillations, their rotational motion decreases, as shown in Fig. 7.1e.

Fig. 7.2

Image adapted with permission from Ref. [17]

a Four transition paths, experimentally observed under a field of amplitude $B_0 = 7.28\,$mT and cone angle $\theta = 26.9^{\circ }$. In all cases the starting frequency is $f = 1\,$Hz. The first image on the top illustrates the Synchronous $\rightarrow$ Asynchronous (SA) transition ($f = 20\,$Hz) with a cell thickness of $h= 5.1\,\mathrm {\mu m}$. The second the Synchronous $\rightarrow$ Exchange ($f = 8\,$Hz) $\rightarrow$ Asynchronous (SEA) transition ($f = 25\,$Hz) $h = 4.4\, \mathrm {\mu m}$, the third, the Synchronous $\rightarrow$ Exchange ($f = 3$ Hz) $\rightarrow$ Rupture (SER) of dimers ($f = 14$ Hz), $h = 4.4\, \mathrm {\mu m}$. The last, the Synchronous $\rightarrow$ Rupture (SR) transition ($f = 9$ Hz), $h = 4\, \mathrm {\mu m}$. b Regions where the transition paths occurs in the $(\Phi , h)$ plane. Scattered symbols are experimental data, shaded regions result from numerical simulations. c Mean rotation frequency versus f and c nearest neighbor separation distance $\langle \Delta r \rangle$ for different cell thickness h and normalized area packing fraction $\Phi$.

The different dynamic states depend on both the frequency f and cell thickness h. As shown in the schematic in Fig. 7.2a, we find four types of transition paths when starting from the synchronous regime, i.e. at low frequency. The first is the Synchronous-Asynchronous (SA), which can be more complex with an intermediate Exchange state (SEA). In such state which is usually observed at high density, the dimers break up, and the composing particles exchange positions by passing close to near particles, forming a rotating dimer for half period of the field, and then breaking again. Further, the synchronous-exchange path may instead be followed by a rupture state (SER), namely the up and down crystal or simply the synchronous state can go directly into the rupture one (SR). The diagram in Fig. 7.2b illustrates their locations when varying the normalized area packing fraction, $\Phi = N\pi d^2/(4A)$, being N the number of particles and A the corresponding area. One can identify these states by measuring two observables related with the particle dynamics. They are the average rotation speed of the dimers, shown in Fig. 7.2c, and $\langle \Delta r \rangle$ which is the average distance between nearest neighbors, Fig. 7.2d. In the numerical work, we focus on the transition between the synchronous-exchange state along the SER path, and carefully analyze how this transition set in by raising f.

3 Numerical Simulation

We perform Brownian dynamics simulations using the free package LAMMPS [24] modified to consider the particle induced dipole moment and an overdamped integrator. In particular, we simulate $N=1000$ paramagnetic colloids with individual positions $\boldsymbol{r}_i=(x_i,y_i,z_i)$ confined in a quasi two dimensional (2D) box of size $L_x \times L_y \times h$.The box has periodic boundary conditions on the $(\boldsymbol{x},\boldsymbol{y})$ plane and fixed walls on the z axis at positions $z=\pm \frac{h}{2}$. For each particle, we integrate the equations of motion,

$$\begin{aligned} \gamma \frac{d \boldsymbol{r}_i}{dt}= \sum _{j\ne i}{\boldsymbol{F}_{int}(\boldsymbol{r}_i-\boldsymbol{r}_j)} + \boldsymbol{F}_{w}+ \boldsymbol{F}_{g} + \boldsymbol{\eta }(t) \end{aligned}$$

(7.4)

where $\gamma$ is the viscous friction, $\boldsymbol{F}_{int}(\boldsymbol{r}_i - \boldsymbol{r}_j)$ is the total force exerted on particle i by particle j, $\boldsymbol{F}_{w}$ is the normal force exerted by the confining walls on particle i, $\boldsymbol{F}_g$ the gravitational force and $\boldsymbol{\eta }(t)$ the force due to the thermal fluctuations. The total force on the particle results from an interaction potential, $\boldsymbol{F}_{int}\left( \boldsymbol{r}_i - \boldsymbol{r}_j\right) = -\nabla U_{int}\left( \boldsymbol{r}_i - \boldsymbol{r}_j\right)$, where $U_{int} \left( \boldsymbol{r}_i - \boldsymbol{r}_j\right) = U_{\textrm{dip}} \left( \boldsymbol{r}_i - \boldsymbol{r}_j\right) + U_{WCA}\left( \left| \boldsymbol{r}_i - \boldsymbol{r}_j\right| \right)$. Here the first term is the magnetic dipolar interaction (Eq. 7.1), while the second one refers to a repulsive Weeks-Chandler-Andersen (WCA) potential $U_{WCA}$, which is given by,

$$\begin{aligned} U_{WCA} = \left\{ \begin{array}{lr} 4\epsilon \left[ \left( \frac{d}{r}\right) ^{12} - \left( \frac{d}{r}\right) ^6 \right] + \epsilon \, \, \, & \text {for } r<2^{\frac{1}{6}}d\\ 0 & \, \, \, \, \, \text {for } r>2^{\frac{1}{6}}d \end{array} \right. \end{aligned}$$

(7.5)

Further, the interaction between the particles and the wall, $U_{w}(z)$ is also given by a WCA potential, with $\boldsymbol{F}_{w}(z) = -\boldsymbol{\nabla } U_{w}(z)$ and the gravitational force is given by $\boldsymbol{F}_g = - \pi \Delta \rho g d^3\boldsymbol{z} /6$ being $\Delta \rho$ the density mismatch between the particles and water and g the gravitational acceleration. Finally we assume that $\boldsymbol{\eta }(t)\equiv (\eta _x,\eta _y,\eta _z)$ are random Gaussian variables with zero mean, $\langle \eta _{i}(t) \rangle =0$ and correlation function: $\langle \eta _i(t)\eta _j(t^\prime )\rangle \equiv 2 k_B T \gamma \delta _{ij} \delta (t-t^\prime )$, being $k_B$ the Boltzmann constant and T the experimental temperature.

In the simulations, we usually fix N and the normalized area fraction $\Phi =\frac{N \pi d^2}{4A}$ being $A=L_x \times L_y = L^2$ and varying the driving frequency f, which is our control parameter. For each value of the frequency, $M=10$ statistically independent runs are performed, each with a randomized initial configuration. The obtained observable are then averaged over these M configurations. We introduce parameters extracted directly from the experiments, such as $\phi =0.262$, $h=3.9 \mu \text {m}$, $B_0=7.28 \,$mT, $d=2.8\, \mu \text {m}$, $\theta =27^{\circ }$, $\gamma = 56.75 \times 10^{-6}\, \mathrm {pN \, s \, nm^{-1}}$, $\Delta \rho = 10^3 \,\mathrm {Kg \, m^{-3}}$, $\chi = 0.4$ and use as value for the WCA potential $\epsilon = 10^4 \,\mathrm {pN \, nm}$ [25, 26]. Our system is initialized by randomly placing the particles in the $z=0$ plane, with the values of the coordinate of each particle center, $x_i$ and $y_i$ being drawn each independently from a uniform distribution in $(-\frac{L}{2},\frac{L}{2})$. We further avoid artifacts due to the close proximity of the particles by imposing that each particle position must be at a minimum distance of $d+\delta$, being $\delta$ a tolerance parameter which is set to $50\, \text {nm}$. Equations 7.4 are then integrated with a simulation time step of $\delta t =10^{-4}s$ and a total simulation time of $t_{tot}= 1000\, s$.

Finally, in the runs that investigate the SER transition path, we switch off gravity to avoid perturbing the final distribution of the up and down particles.

4 The Synchronous-Exchange Transition

4.1 Order Parameter

We use numerical simulations to investigate the SE transition which occurs when the dimers break and exchange particles with their neighbors. The advantage of the simulation is that it allows us to carefully tune the driving frequency and to consider relatively large systems, increasing the statistical average. As an order parameter that allows to distinguish between the synchronous and the exchange state, we use a combination of stroboscopic measurements and Voronoi tessellation, following a previous work [27]. In particular, our procedure is schematically illustrated in Fig. 7.3a–d. First, for a state at time t we define the Voronoi tessellation as the set of polygons $\{ a_i(t)\}$ for $i \in \{ 1,2,...,N\}$ such that the area inside the polygon $a_i(t)$ contains only the points whose (Euclidean) distance is closer to particle i than other particles. Then, a particle is defined as active at time t if its position after half a period, $\boldsymbol{r}_i(t+T/2)$, is contained by its Voronoi polygon half a period before, $a_i(t-T/2)$. The series of images in Fig. 7.3a–d show how this definition applies to the synchronous (a, b) and exchange (c, d) states. In the former regime, the particles within the dimers at $t=t'-T/2$ (a) display similar positions after a period, $t=t'+T/2$ (b), and identical Voronoi tessellation (green mesh). This fact results from the reversible trajectories which cyclically repeat after one period even for isolated particles which in this regime do not perform hopping motion, with negligible deviation due to thermal fluctuations. Thus, all particles in Fig. 7.3, panels (a, b) are considered as passive. The situation changes in the exchange phase, illustrated in panels (c, d). After one driving period, at $t=t'+T/2$ (d), due to the exchange position process, many particles are no longer in their original cells, and this generates active particles, which are highlighted by an orange ring.

Fig. 7.3

a–d Simulation snapshots explaining the classification of particles into active and passive. Top a, b panels corresponds to $f = 2.8\,$Hz while bottom c, d to $f = 4\,$Hz. Left a, c panels are taken at time $t = t' - T /2$, while right panels b, d at $t = t' + T /2$, being T the period of rotation of the magnetic field. The particles colored in red (blue) are close to (far away from) the top wall. The green mesh illustrates the Voronoi tessellation, while the particle trajectories are superimposed to the images in black in panels b, d. The orange disks in panel d indicates colloidal particles that are considered “active”. e Evolution with time of the fraction of active particles $\Psi$ for four different driving frequencies averaged from 10 simulation runs ($N = 1000$). The continuous lines through the data are non linear regressions following a stretched exponential, see text

Once identified the active particles, we define the order parameter $\Psi (t,f)$ as the fraction of such particles at time t; $\Psi (t,f)=N_{active}(t,f)/N$, being N the total number of particles. In Fig. 7.3e we show the time evolution of $\Psi$ averaged over 10 simulations for $N = 1000$ particles, starting from random initial conditions and in the range of frequencies $f\in [3.1, 3.3]\,$Hz. Here it should be noted that single isolated dimers display a transition from synchronous rotation to broken dimer above $3.3\,$Hz, sign of the collective nature of the exchange phase. Moreover, since the range of frequencies is very small, all curves in Fig. 7.3e start from similar initial values. However, after an initial relaxation time, different frequencies produce very different final steady-state values of activity. During the relaxation process, we observe that colloidal particles close enough were able to join forming dimers. At high frequencies, these dimers are not stable but keep breaking and reforming, impeding the system to leave the active state. However, at low frequencies, the dimer can survive long enough to interact repulsively with neighboring dimers and arrange themselves into a low ordered configuration, like that depicted in Fig. 7.3a. In such state, the dimers are mainly stable, and unpaired colloids are isolated, with no possibility of hopping, thus reaching a low activity state.

We find that all curves in Fig. 7.3e can be fitted by a stretched exponential function of the form,

$$\begin{aligned} \Psi (t) = \Psi _0 \exp {\left[ - \left( \frac{t}{\lambda } \right) ^{\beta } \right] } + \Psi _{\infty } \end{aligned}$$

(7.6)

with a stretching exponent $\beta \in [0,1]$. This functional form is commonly used when modeling relaxation in glasses, or to approximate response functions (be it mechanical, electric or magnetic) in disordered media.

4.2 Relaxation Time

Using Eq. 7.6, we could fit all the simulation data, and extract two main parameters, the steady-state activity from the constant $\Psi |_{t \rightarrow \infty }=\Psi _0$ and the relaxation time scale which is given by the first moment of the stretched exponential function [28]:

$$\begin{aligned} \tau _r = \frac{\lambda }{\beta } \Gamma \left( \frac{1}{\beta }\right) \!\! , \end{aligned}$$

(7.7)

being $\Gamma$ the gamma function. Figure 7.4a, b show both quantities as a function of the reduced frequency, here defined as $|f-f_r|/f_r$ being $f_r$ the rupture frequency that bridges the synchronous to the exchange phase. To determine $f_r$, we start by considering the behavior of the asymptotic activity, which is shown in the inset of Fig. 7.4a. While the left of the plot is nearly flat, with a small increase which could be due to thermal fluctuations, $\Psi |_{t \rightarrow \infty }$ rapidly raises for $f>f_r$ following a power law behavior. We fit this plot using a piece-wise function of the form $((f-f_r)/f_r)^{\gamma }$ for $f>f_r$ and 0 otherwise, and extract a value for the critical frequency of $f_r = 3.185 \pm 0.001$ Hz, with an exponent $\gamma = 0.7$. Using the value of $f_r$, we can calculate the reduced frequency as $|f-f_r|/f_r$ and we use it to plot the one branch of the asymptotic activity in Fig. 7.4a. In Fig. 7.4b, we show the relaxation timescale $\tau _r$ extracted from the two branches close to $f_r$ as a function of the reduced frequency, while the small inset shows the full region around $f_r$ in linear scale. Here the right branch ($f > f_r$) is displayed as a solid line, while the left branch ($f < f_r$) as a dotted one. While these data are noisier than the asymptotic activity, they still exhibit a linear behavior in the logarithm plot. We find that both curves are consistent with the 1.23 exponent predicted for the Manna (conserved-DP) universality class [29], which is shown as a dotted line.

Fig. 7.4

a Asymptotic value of the order parameter $\Psi$ in the long time limit against reduced frequency. Here $f_r$ denotes the critical frequency of rupture, calculated from fitting the plot in the inset to a piece-wise power law, which gives an exponent $\gamma =0.7$ (orange lines). b Relaxation time of $\Psi$ after a random initial state. Both branches appear to be consistent with an exponent of 1.23 (dotted line). The inset shows the divergence of the timescale around the critical frequency $f_r$

5 Conclusion

We have investigated the collective dynamics of paramagnetic colloids driven by a precessing field while being confined within two narrow plates. We focus on the transition between two dynamic states, synchronous and exchange, which occurs at large particle densities. We identify a critical rupture frequency that bridges these two states and define an order parameter as the fraction of active particles. We find that the time dependence of the order parameter can be well capture by a stretched exponential function which allows us to measure the relaxation time associated with such transition. We find an exponent consistent with the Manna universality class, which points toward the presence of an absorbing phase transition in such system.

Out of equilibrium phase transitions have been reported in models that describe disparate phenomena, from the onset of turbulence [30], to forest fires [31] and financial crises [32]. Among these, a recurrent observation is the existence of an absorbing phase, which appears in systems that fall into a state from which they can’t get out [33]. Other examples of absorbing phases are observed in models for catalytic chemical reactions [34] interface growth [35], wetting [36], depinning [37], and granular matter [38]. Also recently, periodically sheared emulsions have risen as model systems for non-equilibrium transitions with an absorbing phase; one of the very few experimental realizations of such a transition [39, 40]. We have provided another system that can be used to investigate the fascinating collective physics of these systems when they are driven out of equilibrium by an external field.