Capillary action in scalar active matter

TL;DR

This paper investigates how scalar active matter can exhibit capillary action phenomena similar to classical fluids, driven by emergent effective attractions from particle interactions, with simulations revealing relationships between active sedimentation length and capillary rise.

Contribution

It introduces a minimal model for scalar active matter demonstrating emergent capillarity due to purely repulsive interactions, a phenomenon not observed in passive fluids.

Findings

Capillary rise is proportional to active sedimentation length $\\lambda$.

Wetting height grows superlinearly with $\\lambda$.

Imbibition height scales with sedimentation length and packing fraction.

Abstract

We study the capacity of active matter to rise in thin tubes against gravity and other related phenomena, like, wetting of vertical plates and spontaneous imbibition, where a wetting liquid is drawn into a porous medium. This capillary action or capillarity is well known in classical fluids and originates from attractive interactions between the liquid molecules and the container walls, and from the attraction of the liquid molecules among each other. We observe capillarity in a minimal model for scalar active matter with purely repulsive interactions, where an effective attraction emerges due to slowdown during collisions between active particles and between active particles and walls. Simulations indicate that the capillary rise in thin tubes is approximately proportional to the active sedimentation length $\lambda$ and that the wetting height of a vertical plate grows superlinear with $\lambda$. In a disordered porous medium the imbibition height scales as $\langle h\rangle\propto\lambda\phi_m$, where $\phi_m$ is its packing fraction.

Equations

$$\lambda \propto \frac{V_0^2}{\alpha V_g}$$

$$\mathbf{m}=\left(\begin{array}{c}{m}_x\\ m_y\end{array}\right)=\frac{1}{\rho }\left(\begin{array}{c}{\rho }_r-{\rho }_l\\ {\rho }_u-{\rho }_d\end{array}\right)$$

$$\Delta h\propto {\lambda }^{0.9},$$

$$\Delta H\propto {\lambda }^{1.3}$$

$$⟨h⟩\propto \lambda {\varphi }_m$$

Full text

Introduction – Recently, active matter, which consists of particles (motile microorganism or active colloids) that consume nutrients or fuel and convert it into a persistent motion, has received a lot of attention due to its intrinsic out-of-equilibrium character on the microscale Marchetti et al. (2013); Elgeti et al. (2015). The simplest representative of active matter are spherically symmetric, active Brownian particles (ABPs) without alignment, however, with excluded volume interactions Fily and Marchetti (2012); Redner et al. (2013a). Further representatives of the same class, also called scalar active matter, are, for example, run-and-tumble particles Tailleur and Cates (2008); Solon et al. (2015a) and active lattice gas Thompson et al. (2011); Sepúlveda and Soto (2017); Whitelam et al. (2018); Kourbane-Houssene et al. (2018). Such system, although far from equilibrium, are in some sense reminiscent of a passive fluid with attractive interactions Cates and Tailleur (2015); Farage et al. (2015), since ABPs slow down during collisions and effectively attract each other. As a result, ABPs undergo a motility-induced phase separation (MIPS) into a coexisting dense and dilute phase Fily and Marchetti (2012); Redner et al. (2013a); Stenhammar et al. (2014); Wysocki et al. (2014). The same mechanism is responsible for adhesion of ABPs to repulsive walls, an effect called wall accumulation Lee (2013); Elgeti and Gompper (2013); Sepúlveda and Soto (2017).

The investigation of capillary action, the ability of liquids to rise in thin tubes against the action of gravity, has a long history and goes back to Leonardo da Vinci De Gennes et al. (2013). It originates from the attractive interactions between the liquid molecules and the container walls and from the attraction of the liquid molecules among each other (causing surface tension). The height of the liquid column in the tube is governed by the balance between the gain in surface energy, which is proportional to the difference of the solid–vapor and the solid–liquid surface tension, and the cost in gravitational energy De Gennes et al. (2013); Rowlinson and Widom (2013). The classical picture seems to prohibit the appearance of capillarity in systems with purely repulsive interactions, however, as mentioned above, scalar active matter display in some sense an equilibrium behavior Fodor et al. (2016) of an attractive fluid with an equation of state for the pressure and equality of pressures in coexisting phases Solon et al. (2015b, c); Das et al. (2019). Still, their interfacial properties are contradicting: despite stable liquid-gas interfaces the surface tension was found to be negative Bialké et al. (2015); Patch et al. (2018); Solon et al. (2018), which is also true for solid-liquid interfaces Zakine et al. (2019). All in all, the appearance of capillarity and related phenomena in active systems is not trivial and was not investigated so far.

The behaviour of active particles in a homogeneous medium or near flat surfaces is well-studied. However, the natural habitat for many motile bacteria are porous media, such as soil, tissue, or biofilm. So far the transport properties of interacting and non-interacting active particles in heterogeneous media under both quiescent and flow conditions Chepizhko et al. (2013); Reichhardt and Olson Reichhardt (2014); Pinçe et al. (2016); Alonso-Matilla et al. (2019) have been studied. To the best of our knowledge, the classical experiment of spontaneous imbibition of a liquid into a porous medium Alava et al. (2004); Gruener et al. (2012) has not been performed in the context of active fluids up to now.

Model – Several minimal off-lattice models of isotropic active particles without alignment interaction have been proposed, e.g., active Brownian particles Fily and Marchetti (2012), run-and-tumble particles Solon et al. (2015a) or active Ornstein-Uhlenbeck particles Farage et al. (2015). By contrast, we use a lattice model of scalar active matter, the so-called active lattice gas (ALG) Thompson et al. (2011); Sepúlveda and Soto (2017); Whitelam et al. (2018); Kourbane-Houssene et al. (2018), which allows for simulation of large systems and, in addition, can be described by exact hydrodynamic equations on macroscopic scales Kourbane-Houssene et al. (2018).

We consider $N$ particles on a square lattice with $N_{x}\times N_{y}=(nL_{x})\times(nL_{y})$ sites, where $1/n$ corresponds to grid spacing, as will be explained later. Four types of particles $\sigma_{i_{x},i_{y}}\in\{l,r,u,d\}$ corresponding to particles in a left, right, up and down drifting state can occupy a lattice site $(i_{x},i_{y})$, c.f. the sketch in Fig. S1 in Supplemental Material SI . In a simple exclusion process the maximum occupation number per site is $1$. We label immobile obstacle sites as $\sigma_{i_{x},i_{y}}=w$ and use them in order to construct walls, tubes and porous matrices. An empty site is indicated by $\sigma_{i_{x},i_{y}}=0$. ALG evolves according to four possible processes, see Supplemental Material SI for the corresponding microscopic equations for the evolution of the average occupation number:

1. Symmetric diffusion: For nearest neighbor pairs $(i_{x},i_{y})$ and $(j_{x},j_{y})$, $\sigma_{i_{x},i_{y}}\neq w$ and $\sigma_{j_{x},j_{y}}\neq w$ are exchanged at rate $D$.
2. Self-propulsion: $\forall(i_{x},i_{y})$: a particle of type $\sigma_{i_{x},i_{y}}=l$ in $(i_{x},i_{y})$ jumps to the left to $(i_{x}-1,i_{y})$ if $\sigma_{i_{x}-1,i_{y}}=0$ at rate $V_{0}/n$ and analogous for particles of type $r$, $u$ and $d$.
3. Sedimentation: $\forall(i_{x},i_{y})$: particles of all types in $(i_{x},i_{y})$ jumps downwards to $(i_{x},i_{y}-1)$ if $\sigma_{i_{x},i_{y}-1}=0$ at rate $V_{g}/n$.
4. Tumbling: Particle switch state at rate $\alpha/n^{2}$.

Such rules generate a persistent random walk Bertrand et al. (2018); Whitelam et al. (2018) and the model exhibits motility-induced phase separation about a critical activity and density as a result of excluded volume interactions Kourbane-Houssene et al. (2018); Whitelam et al. (2018). We simulate the dynamics of the ALG via a random-sequential update together with a rejection-free continuous-time Monte Carlo algorithm Bortz et al. (1975). In detail, the update sequence is a randomly permuted set of the particle labels, a process is chosen with probability $W/R$ and the time increment after each move is $1/(NR)$, where $W$ denotes the rate of the chosen process and $R=4D+V_{0}/n+V_{g}/n+3\alpha/n^{2}$ is the total rate.

With the above choice of rates and after diffusive rescaling of space $x=i/n$ and time $\tau=t/n^{2}$ all processes contribute equally to the macroscopic description of the system De Masi et al. (1985). Hydrodynamic equations for the densities of the four species $\{l,r,u,d\}$ can be derived in case of large $n$ with diffusion constant $D$, tumbling rate $\alpha$, propulsion $V_{0}$ and sedimentation velocity $V_{g}$, see Eq. (17-20), in Supplemental Material SI .

Important dimensionless numbers are the active $Pe_{a}=V_{0}/\sqrt{D\alpha}$ and gravitational Péclet number $Pe_{g}=V_{g}/\sqrt{D\alpha}$, which compare either active swimming or gravitation-induced drift motion to thermal diffusion. We use $l=\sqrt{D/\alpha}$ as a length scale.

Our basic setup consists of a ALG confined between horizontal walls at $y=0$ and $y=L_{y}$, we apply periodic boundary conditions along $x$-direction and gravity acts along the negative $y$-direction. As a consequence, a dense phase covers the lower wall with a dilute phase on top of it. The density profile of the dilute phase decays exponentially as $\rho(y)\propto\exp{(-y/\lambda)}$, where $\lambda$ is the active sedimentation length, which scales as

[TABLE]

for large activity $Pe_{a}$ Tailleur and Cates (2009); Enculescu and Stark (2011); Solon et al. (2015a); Ginot et al. (2018), as have been confirmed by our simulations. In the absence of gravity active particles would accumulate symmetrically at both walls and in case of an ideal ALG at large $Pe_{a}$ the density profile, for instance, at the lower wall $y=0$ would decay as $\rho(y)\propto C_{1}\exp{(-y/\lambda_{1})}+C_{2}\exp{(-y/\lambda_{2})}$ with $\lambda_{1}=D/V_{0}$ and $\lambda_{2}=\sqrt{D/2\alpha}$, which can be calculated exactly using the corresponding hydrodynamic equations SI .

We insert a capillary or a porous matrix into the dense phase (or confine the system along $x$-direction by vertical walls), let the system evolve until steady state is reached and adjust simultaneously the number of particles in order to fix the height of the interface between the dense and the dilute phase in the bulk region (far away from perturbations due to vertical walls or porous matrix). We define the position of the interface as the isodensity curve $\rho(x,y)=0.6$, where $\rho=\rho_{l}+\rho_{r}+\rho_{u}+\rho_{d}$ is the total density, and fix the height of the interface in the bulk region in all simulations to $y_{bulk}/l=4$. In the following, we measure heights, e.g., the height of the meniscus $\Delta h$ in case of capillary rise, relative to $y_{bulk}$. We choose the size of the system, such that $L_{x}$ and $L_{y}$ are much larger than any other length scale in system, like swimming persistence length $V_{0}/\alpha$ or active sedimentation length $\lambda$. A bulk ALG (no walls) does not phase separate below a critical value $Pe_{a}^{c}=8$ Mangeat (2019).

Because the results of the Monte Carlo simulations with a grid resolution of $n=10$ match results with $n=20$ and are consistent with the numerical solutions, obtained using finite element method Hecht (2012), of the hydrodynamic equations SI corresponding to the limit $n\rightarrow\infty$, we used a resolution of $n=10$ in the following.

Capillary rise – In order to study capillarity in active systems we use our basic setup and insert a tube into the dense phase. A typical result for a thin tube and at sufficiently high $Pe_{a}$ and small $Pe_{g}$ is shown in Fig. 1 with the total density $\rho(x,y)\in[0,1]$ in Fig. 1(a), the absolute value of the normalized polarization field

[TABLE]

in Fig. 1(b) and the phase $\varphi(x,y)\in[-\pi,\pi]$ of $\mathbf{m}=|\mathbf{m}|\cdot(\cos{\varphi},\sin{\varphi})$ in Fig. 1(c), which indicates the direction of the polarization $\mathbf{m}$. The ALG wets the walls of the tube accompanied by a strong polarization in the direction antiparallel to the surface normal, the dense phase fills the tube about the level of the bulk and a concave meniscus develops within the tube. Accumulation at boundaries Elgeti and Gompper (2013); Sepúlveda and Soto (2017); Ostapenko et al. (2018) and capillary condensation in slit pores Ni et al. (2015); Wittmann and Brader (2016) are a well-known effect in scalar active matter, what is new, is the rise of the dense phase inside the tube against gravity.

Fig. 2(a,b) illustrates the dependence of the height of the meniscus $\Delta h$ on activity $Pe_{a}$ and gravity $Pe_{g}$ at fixed capillary width $\delta x/l=1$. As intuitively expected, $\Delta h$ increases with activity and decreases with gravity. Moreover, $\Delta h$ increases strongly with the tube height $\delta y$ till some critical value and saturates above it. In the latter regime we observe an approximate scaling $\Delta h\propto V_{0}^{1.8}$ and $\Delta h\propto V_{g}^{-0.85}$. For every parameter pair $(Pe_{a},Pe_{g})$ we estimate the corresponding active gravitational length $\lambda$ from a fit of the density profile $\rho(y)$ in the bulk (far away from the tube) to $\rho(y)\propto\exp{(-y/\lambda)}$. Using this, we obtain a master curve from data shown in Fig. 2(a,b) by plotting $\Delta h$ as a function of $\lambda$ and rescaling $\Delta h$ for different tube heights by $\delta y^{0.24}$. As a result, we get a simple scaling relation

[TABLE]

as indicated in Fig. 2(c).

The height of the meniscus $\Delta h$ decreases strongly with increasing capillary width $\delta x$, see Fig. 2(d), and the curves for different parameter pairs $(Pe_{a},Pe_{g})$ collapse after rescaling of $\Delta h$ with $\lambda^{0.9}$, which is consistent with the master curve $\Delta h\propto\lambda^{0.9}$ in Fig. 2(c). For large $Pe_{a}$ and $\delta x$ the data collapse is not perfect, where $\Delta h$ even becomes negative.

Wetting of a vertical plate – As mentioned above, the dense phase of the ALG not only rises inside a capillary but also wets the outside walls of the capillary. In order to study the latter effect separately we consider a setup of a rectangular container with horizontal walls at $y=0$ and $y=L_{y}$ and vertical walls at $x=0$ and $x=L_{x}$. Again we adjust the number of particles in order to fix the height of the interface far away from walls ($x=L_{x}/2$). A typical total density field $\rho(x,y)\in[0,1]$ together with the isoline at $\rho=0.6$ (definition of the interface) is shown in Fig. 3(a), where a pronounced wetting of the vertical wall is visible. We estimate the height of the wetting layer $\Delta H$ for different parameter pairs $(Pe_{a},Pe_{g})$ together with the corresponding active gravitational lengths $\lambda$ and obtaine a master curve, which suggests a superlinear growth

[TABLE]

of the wetting height. Wetting can be understood as follows, activity leads to an accumulation of particles at the container walls and gravity forces particles to sediment to the bottom, however, excluded volume effect prevent a collapse of the wetting layer. A zero-order approximation for the interface profile in a capillary tube of a width $\delta x$ ia a superposition of wetting profiles of two independent walls at distance $\delta x$.

Imbibition of a porous matrix – Motivated by above results we probe how the ALG penetrates an porous medium. We construct a porous matrix from randomly placed non-overlapping hard discs with uniformly distributed diameters in the range $\sigma/l\in[0.5,2]$ and a minimum gap size of $0.325l$. As in the previous simulation setups, we adjust the number of particles in order to fix the interface far away from the matrix. An example of a spontaneous imbibition is shown in Fig. 4 for porosity (or void fraction) of $1-\phi_{m}=0.2$ corresponding to a packing fraction of the porous matrix of $\phi_{m}=0.8$. After an initial rise the invasion front reaches a final height and width, and the interface fluctuates slightly in the steady state.

An obvious question is: What is the effect of the porosity on the imbibition? In Fig. 5(a) we present the average density profiles $\rho(y)$ of the active lattice gas within the porous matrix for different porosities $1-\phi_{m}$ at $(Pe_{a},Pe_{g})=(15,1)$ and indicate the position of the interface by circles; the horizontal average was performed over the void region only and over 10 independent realizations of the matrix. As expected, the smaller the porosity the higher the rise of the dense phase within the porous matrix. Interestingly, the density of the dilute phase above the invasion front within the matrix is significantly larger then the bulk density at the same height, see dashed line in Fig. 5(a).

We perform similar simulations for different $(Pe_{a},Pe_{g})$ and estimate the mean interface height $\langle h\rangle$ and the corresponding active gravitational length $\lambda$. A scaling of $\langle h\rangle$ with $\lambda$ leads to a reasonable data collapse and indicates a simple growth relation

[TABLE]

of the interface height, see Fig. 5(b). A rough comparison between imbibition and capillary rise can be done by plotting together the mean interface height $\langle h\rangle$ versus mean gaps size of the porous matrix, obtained with Delaunay triangulation, and the height of the meniscus $\Delta h$ versus capillary width $\delta x$, see Fig. 2(d). Although the shape of the curves is similar, the capacity of ALG to rise in porous media against the action of gravity is twice as strong as in thin tubes.

Conclusions and Outlook – We have show that, scalar active matter with purely repulsive interactions can rise in thin tubes or invade porous matrix against gravity. As a proof of concept we have used one of the simplest active matter models and there are several obvious extensions and future directions in the study of active capillarity and related phenomena using more realistic and elaborated models. It is obvious to consider next off-lattice models, such as active Brownian particles (ABPs), in order to test the universality of the scaling relations; preliminary simulations clearly demonstrate capillary action in ABPs and confirm the robustness of the effect in active matter systems. Particle-particle and particle-wall hydrodynamic interactions Drescher et al. (2011); Kantsler et al. (2013) or, in case of anisotropic active particles, alignment interactions can have a dramatic effect on the behaviour near boundaries as compared to scalar active matter and should be taken into account in future models. It would be very interesting to investigate the modification of classical capillary rise Dimitrov et al. (2007) by activity using active colloids with attraction Redner et al. (2013b), where it is predicted that wall attraction may cause capillary drying Wittmann and Brader (2016). Also the study of capillary action and wetting in active multiphase system, such as suspensions of motile bacteria Lauga (2016), algae Goldstein (2015) or synthetic self-propelled particles Palacci et al. (2013); Buttinoni et al. (2013), is very promising and could be a possible experimental test for our predictions.