Introduction¶
In this tutorial we explore the ways to simulate self-propulsion in the simulation software package ESPResSo. We consider three examples that illustrate the properties of these systems. First, we study the concept of enhanced diffusion of a self-propelled particle. Second, we investigate rectification in an asymmetric geometry. Finally, we determine the flow field around a self-propelled particle using lattice-Boltzmann simulations (LB). These three subsections should give insight into the basics of simulating active matter with ESPResSo. The tutorial assumes basic knowledge of Python and ESPResSo, as well as the use of lattice-Boltzmann within ESPResSo. It is therefore recommended to go through the relevant tutorials first, before attempting this one.
Active particles¶
Active matter is a term that describes a class of systems, in which agents (particles) constantly consume energy to perform work, for example in the form of directed motion. These systems are therefore highly out-of-equilibrium and thus defy description using the standard framework of statistical mechanics. Active systems are, however, ubiquitous. On our length scale, we encounter flocks of birds, schools of fish, and, of course, humans; on the mesoscopic level examples are found in bacteria, sperm, and algae; and on the nanoscopic level, transport along the cytoskeleton is achieved by myosin motors.
Active Particles in ESPResSo¶
The ENGINE feature offers intuitive syntax for adding self-propulsion to a particle. Propulsion will occur along the vector that defines the orientation of the particle (henceforth referred to as ‘director’). Within the ENGINE feature there are two ways of setting up a self-propelled particle, with and without coupling to a fluid.
Self-Propulsion without Hydrodynamics¶
For this type of self-propulsion the Langevin thermostat can be used. The
Langevin thermostat imposes a velocity-dependent friction on a particle.
When a constant force is applied along the director,
the friction causes the particle to attain a terminal velocity, due to the balance
of driving and friction force, see Fig. 1. The timescale on
which the particle's velocity relaxes towards this value depends on the
strength of the friction and the mass of the particle. The ENGINE
feature implies that rotation of the particles (the ROTATION feature) is
compiled into ESPResSo. The particle can thus reorient due to external torques or
due to thermal fluctuations, whenever the rotational degrees of freedom are
thermalized. Note that the rotation of the particles has to be enabled
explicitly via their rotation property. The ‘engine’ building block can
be connected to other particles, e.g., via the virtual sites (rigid
body) to construct complex self-propelled objects.
Enhanced Diffusion¶
First we import the necessary modules, define the parameters and set up the system.
%matplotlib inline
import matplotlib.pyplot as plt
plt.rcParams.update({'font.size': 18})
import tqdm
import numpy as np
import espressomd.observables
import espressomd.accumulators
espressomd.assert_features(
["ENGINE", "ROTATION", "MASS", "ROTATIONAL_INERTIA", "CUDA"])
ED_PARAMS = {'time_step': 0.01,
'box_l': 3*[10.],
'skin': 0.4,
'active_velocity': 5,
'kT': 1,
'gamma': 1,
'gamma_rotation': 1,
'mass': 0.1,
'rinertia': 3*[1.],
'corr_tmax': 100}
ED_N_SAMPLING_STEPS = 5000000
system = espressomd.System(box_l=ED_PARAMS['box_l'])
system.cell_system.skin = ED_PARAMS['skin']
system.time_step = ED_PARAMS['time_step']
Exercise¶
- Use
ED_PARAMSto set up a Langevin thermostat for translation and rotation of the particles.
system.thermostat.set_langevin(kT=ED_PARAMS['kT'],
gamma=ED_PARAMS['gamma'],
gamma_rotation=ED_PARAMS['gamma_rotation'],
seed=42)
The configuration for the Langevin-based swimming is exposed as an attribute of the ParticleHandle class of ESPResSo, which represents a particle in the simulation. You can either set up the self-propulsion during the creation of a particle or at a later stage.
part_act = system.part.add(pos=[5.0, 5.0, 5.0], swimming={'v_swim': ED_PARAMS['active_velocity']},
mass=ED_PARAMS['mass'], rotation=3 * [True], rinertia=ED_PARAMS['rinertia'])
part_pass = system.part.add(pos=[5.0, 5.0, 5.0],
mass=ED_PARAMS['mass'], rotation=3 * [True], rinertia=ED_PARAMS['rinertia'])
Next we set up three ESPResSo correlators for the Mean Square Displacement (MSD), Velocity Autocorrelation Function (VACF) and the Angular Velocity Autocorrelation Function (AVACF).
pos_obs = espressomd.observables.ParticlePositions(
ids=[part_act.id, part_pass.id])
msd = espressomd.accumulators.Correlator(obs1=pos_obs,
corr_operation="square_distance_componentwise",
delta_N=1,
tau_max=ED_PARAMS['corr_tmax'],
tau_lin=16)
system.auto_update_accumulators.add(msd)
vel_obs = espressomd.observables.ParticleVelocities(
ids=[part_act.id, part_pass.id])
vacf = espressomd.accumulators.Correlator(obs1=vel_obs,
corr_operation="componentwise_product",
delta_N=1,
tau_max=ED_PARAMS['corr_tmax'],
tau_lin=16)
system.auto_update_accumulators.add(vacf)
ang_obs = espressomd.observables.ParticleAngularVelocities(
ids=[part_act.id, part_pass.id])
avacf = espressomd.accumulators.Correlator(obs1=ang_obs,
corr_operation="componentwise_product",
delta_N=1,
tau_max=ED_PARAMS['corr_tmax'],
tau_lin=16)
system.auto_update_accumulators.add(avacf)
No more setup needed! We can run the simulation and plot our observables.
for i in tqdm.tqdm(range(100)):
system.integrator.run(int(ED_N_SAMPLING_STEPS/100))
system.auto_update_accumulators.remove(msd)
msd.finalize()
system.auto_update_accumulators.remove(vacf)
vacf.finalize()
system.auto_update_accumulators.remove(avacf)
avacf.finalize()
taus_msd = msd.lag_times()
msd_result = msd.result()
msd_result = np.sum(msd_result, axis=2)
taus_vacf = vacf.lag_times()
vacf_result = np.sum(vacf.result(), axis=2)
taus_avacf = avacf.lag_times()
avacf_result = np.sum(avacf.result(), axis=2)
fig_msd = plt.figure(figsize=(10, 6))
plt.plot(taus_msd, msd_result[:, 0], label='active')
plt.plot(taus_msd, msd_result[:, 1], label='passive')
plt.xlim((taus_msd[1], None))
plt.loglog()
plt.xlabel('t')
plt.ylabel('MSD(t)')
plt.legend()
plt.show()
The Mean Square Displacement of an active particle is characterized by a longer ballistic regime and an increased diffusion coefficient for longer lag times. In the overdamped limit it is given by
$$ \langle r^{2}(t) \rangle = 6 D t + \frac{v^{2} \tau^{2}_{R}}{2} \left[ \frac{2 t}{\tau^{2}_{R}} + \exp\left( \frac{-2t}{\tau^{2}_{R}} \right) - 1 \right], $$where $\tau_{R} = \frac{8\pi\eta R^{3}}{k_{B} T}$ is the characteristic time scale for rotational diffusion and $ D = \frac{k_B T}{\gamma}$ is the translational diffusion coefficient.
For small times ($t \ll \tau_{R}$) the motion is ballistic
$$\langle r^{2}(t) \rangle = 6 D t + v^{2} t^{2},$$while for long times ($t \gg \tau_{R}$) the motion is diffusive
$$\langle r^{2}(t) \rangle = (6 D + v^{2}\tau_{R}) t.$$Note that no matter the strength of the activity, provided it is some finite value, the crossover between ballistic motion and enhanced diffusion is controlled by the rotational diffusion time.
The passive particle also displays a crossover from a ballistic to a diffusive motion. However, the crossover time $\tau_{C}=\frac{m}{\gamma}$ is not determined by the rotational motion but instead by the mass of the particles.
From the longterm MSD of the active particles we can define an effective diffusion coefficient $D_{\mathrm{eff}} = D + v^{2}\tau_{R}/6$. One could be tempted to also connect this increased diffusion with an effective temperature. However, this apparent equivalence leads to problems when one then attempts to apply statistical mechanics to such systems at the effective temperature. That is, there is typically more to being out-of-equilibrium than can be captured by a simple remapping of equilibrium parameters, as we will see in the second part of the tutorial.
From the autocorrelation functions of the velocity and the angular velocity we can see that the activity does not influence the rotational diffusion. Yet the directed motion for $t<\tau_{R}$ leads to an enhanced correlation of the velocity.
def acf_stable_regime(x, y):
"""
Remove the noisy tail in autocorrelation functions of finite time series.
"""
cut = np.argmax(y <= 0.) - 2
assert cut >= 1
return (x[1:cut], y[1:cut])
fig_vacf = plt.figure(figsize=(10, 6))
plt.plot(*acf_stable_regime(taus_vacf, vacf_result[:, 0]), label='active')
plt.plot(*acf_stable_regime(taus_vacf, vacf_result[:, 1]), label='passive')
plt.xlim((taus_vacf[1], None))
plt.loglog()
plt.xlabel('t')
plt.ylabel('VACF(t)')
plt.legend()
plt.show()
fig_avacf = plt.figure(figsize=(10, 6))
plt.plot(*acf_stable_regime(taus_avacf, avacf_result[:, 0]), label='active')
plt.plot(*acf_stable_regime(taus_avacf, avacf_result[:, 1]), label='passive')
plt.xlim((taus_avacf[1], None))
plt.loglog()
plt.xlabel('t')
plt.ylabel('AVACF(t)')
plt.legend()
plt.show()
Before we go to the second part, it is important to clear the state of the system.
def clear_system(system):
system.part.clear()
system.thermostat.turn_off()
system.constraints.clear()
system.auto_update_accumulators.clear()
system.time = 0.
clear_system(system)
Rectification¶
In the second part of this tutorial you will consider the ‘rectifying’ properties of certain asymmetric geometries on active systems. Rectification can be best understood by considering a system of passive particles first. In an equilibrium system where noninteracting particles are confined to any volume with hard walls as the only potential energy contribution, we know that the particle density is homogeneous throughout the whole available space according to the Boltzmann factor $e^{U(x)/k_BT}$. In an out-of-equilibrium setting however, asymmetric boundaries can lead to a heterogeneous distribution of particles in the steady state.
The geometry we will use is a cylindrical system with a funnel dividing two halves of the box as shown in Fig. 2.
import espressomd.shapes
import espressomd.math
RECT_PARAMS = {'length': 100,
'radius': 20,
'funnel_inner_radius': 3,
'funnel_angle': np.pi / 4.0,
'funnel_thickness': 0.1,
'n_particles': 500,
'active_velocity': 7,
'time_step': 0.01,
'wca_sigma': 0.5,
'wca_epsilon': 0.1,
'skin': 0.4,
'kT': 0.1,
'gamma': 1.,
'gamma_rotation': 1}
RECT_STEPS_PER_SAMPLE = 100
RECT_N_SAMPLES = 500
TYPES = {'particles': 0,
'boundaries': 1}
box_l = np.array(
[RECT_PARAMS['length'], 2*RECT_PARAMS['radius'], 2*RECT_PARAMS['radius']])
system.box_l = box_l
system.cell_system.skin = RECT_PARAMS['skin']
system.time_step = RECT_PARAMS['time_step']
system.thermostat.set_langevin(
kT=RECT_PARAMS['kT'], gamma=RECT_PARAMS['gamma'], gamma_rotation=RECT_PARAMS['gamma_rotation'], seed=42)
cylinder = espressomd.shapes.Cylinder(
center=0.5 * box_l,
axis=[1, 0, 0], radius=RECT_PARAMS['radius'], length=RECT_PARAMS['length'], direction=-1)
system.constraints.add(shape=cylinder, particle_type=TYPES['boundaries'])
# Setup walls
wall = espressomd.shapes.Wall(dist=0, normal=[1, 0, 0])
system.constraints.add(shape=wall, particle_type=TYPES['boundaries'])
wall = espressomd.shapes.Wall(dist=-RECT_PARAMS['length'], normal=[-1, 0, 0])
system.constraints.add(shape=wall, particle_type=TYPES['boundaries'])
funnel_length = (RECT_PARAMS['radius']-RECT_PARAMS['funnel_inner_radius']
)/np.tan(RECT_PARAMS['funnel_angle'])
Exercise¶
- Using
funnel_lengthand the geometric parameters inRECT_PARAMS, set up the funnel cone (Hint: Use a Conical Frustum)
ctp = espressomd.math.CylindricalTransformationParameters(
axis=[1, 0, 0], center=box_l/2.)
hollow_cone = espressomd.shapes.HollowConicalFrustum(
cyl_transform_params=ctp,
r1=RECT_PARAMS['funnel_inner_radius'], r2=RECT_PARAMS['radius'],
thickness=RECT_PARAMS['funnel_thickness'],
length=funnel_length,
direction=1)
system.constraints.add(shape=hollow_cone, particle_type=TYPES['boundaries'])
Exercise¶
- Set up a WCA potential between the walls and the particles using the parameters in
RECT_PARAMS
system.non_bonded_inter[TYPES['particles'], TYPES['boundaries']].wca.set_params(
epsilon=RECT_PARAMS['wca_epsilon'], sigma=RECT_PARAMS['wca_sigma'])
Exercise¶
- Place an equal number of swimming particles (the total number should be
RECT_PARAMS['n_particles']) in the left and the right part of the box such that the center of mass is exactly in the middle. (Hint: Particles do not interact so you can put multiple in the same position) - Particles must be created with a random orientation
for i in range(RECT_PARAMS['n_particles']):
pos = box_l / 2
pos[0] += (-1)**i * 0.25 * RECT_PARAMS['length']
# https://mathworld.wolfram.com/SpherePointPicking.html
theta = np.arccos(2. * np.random.random() - 1)
phi = 2. * np.pi * np.random.random()
director = [np.sin(theta) * np.cos(phi),
np.sin(theta) * np.cos(phi),
np.cos(theta)]
system.part.add(pos=pos, swimming={'v_swim': RECT_PARAMS['active_velocity']},
director=director, rotation=3*[True])
com_deviations = list()
times = list()
Exercise¶
- Run the simulation using
RECT_N_SAMPLESandRECT_STEPS_PER_SAMPLEand calculate the deviation of the center of mass from the center of the box in each sample step. (Hint: Center of mass) - Save the result and the corresponding time of the system in the lists given above.
for _ in tqdm.tqdm(range(RECT_N_SAMPLES)):
system.integrator.run(RECT_STEPS_PER_SAMPLE)
com_deviations.append(system.galilei.system_CMS()[0] - 0.5 * box_l[0])
times.append(system.time)
def moving_average(data, window_size):
return np.convolve(data, np.ones(window_size), 'same') / window_size
smoothing_window = 10
com_smoothed = moving_average(com_deviations, smoothing_window)
fig_rect = plt.figure(figsize=(10, 6))
plt.plot(times[smoothing_window:-smoothing_window],
com_smoothed[smoothing_window:-smoothing_window])
plt.xlabel('t')
plt.ylabel('center of mass deviation')
plt.show()
Even though the potential energy inside the geometry is 0 in every part of the accessible region, the active particles are clearly not Boltzmann distributed (homogenous density). Instead, they get funneled into the right half, showing the inapplicability of equilibrium statistical mechanics.
Hydrodynamics of self-propelled particles¶
In this final part of the tutorial we simulate and visualize the flow field around a self-propelled swimmer.
Of particular importance for self-propulsion at low Reynolds number is the fact that active systems are net force free. This is because the particles have to generate their propulsion on their own as opposed to being dragged by an external force. As a consequence, the flow field around a swimmer does not contain a monopolar (Stokeslet) contribution. In the case of an E. Coli cell, see Fig. 3(a), the reasoning is as follows: The corkscrew tail pushes against the fluid and the fluid pushes against the tail, which generates the forward motion. At the same time the head experiences drag, pushing against the fluid and being pushed back against by the fluid. In total, both the swimmer and the fluid experience no net force even though the swimmer moves. When there is no net force on the fluid, the lowest-order multipole that can be present is a hydrodynamic dipole. Since a dipole has an orientation, there are two basic types of swimmer: pushers and pullers. The distinction is made by whether the particle pulls fluid in from the front (puller, Fig. 3(b,d)) or pushes fluid away from the back (pusher, Fig. 3(a,c)).
In situations where hydrodynamic interactions between swimmers or swimmers and objects are of importance, we use lattice-Boltzmann (LB) to simulate the fluid. We recommend the GPU-based variant of LB in ESPResSo, since it is much faster. Moreover, the current implementation of the CPU self-propulsion with hydrodynamics is limited to one core.
ESPResSo implements swimming coupled with hydrodynamics by applying a force to the particle along its director as in the 'dry' case above, but now also applying a counterforce to the fluid that implicitly models the propulsion mechanism (flagella, cilia, etc).
The friction between the particle and the fluid then results in net force-free swimming.
For the setup of swimming particles with hydrodynamics we cannot use the v_swim argument anymore because it is not trivial to determine the friction acting on the particle. Instead, we have to provide the keys f_swim and dipole_length. Together they determine the dipole strength and the terminal velocity of the swimmer.
The keyword mode (possible values: "pusher", "puller") determines where the counterforce on the fluid is applied, see again
Fig. 3(c, d).
Caveats of the algorithm are:
- One should make sure that the
dipole_lengthis at least one lattice Boltzmann grid spacing, since velocities are interpolated from that grid. If the length is less than one grid spacing, you can easily run into discretization artifacts or cause the particle not to move. - The stability of the swimming algorithm is governed by the stability limitations of the LB method.
- Since the particle is essentially a point particle, there is no rotation caused by shear flow, e.g., a swimmer in a Poiseuille flow. If the thermostat is switched on, the rotational degrees of freedom will be thermalized, but there is no contribution of rotation due to ‘external’ flow fields. It is recommended to use an alternative means of obtaining rotations in your LB swimming simulations. For example, by constructing a raspberry particle.
import espressomd.lb
clear_system(system)
HYDRO_PARAMS = {'box_l': 3*[35],
'time_step': 0.01,
'skin': 1,
'agrid': 1,
'dens': 0.1,
'visc': 5,
'gamma': 2,
'mass': 1,
'dipole_length': 4,
'active_force': 0.01,
'mode': 'pusher'}
HYDRO_N_STEPS = 2000
system.box_l = HYDRO_PARAMS['box_l']
system.cell_system.skin = HYDRO_PARAMS['skin']
system.time_step = HYDRO_PARAMS['time_step']
system.min_global_cut = HYDRO_PARAMS['dipole_length']
Exercise¶
- Using
HYDRO_PARAMS, set up a lattice-Boltzmann fluid and activate it as a thermostat (Hint: lattice-Boltzmann)
lbf = espressomd.lb.LBFluidGPU(agrid=HYDRO_PARAMS['agrid'], dens=HYDRO_PARAMS['dens'],
visc=HYDRO_PARAMS['visc'], tau=HYDRO_PARAMS['time_step'])
system.actors.add(lbf)
system.thermostat.set_lb(LB_fluid=lbf, gamma=HYDRO_PARAMS['gamma'], seed=42)
box_l = np.array(HYDRO_PARAMS['box_l'])
pos = box_l/2.
pos[2] = -15.
Exercise¶
- Using
HYDRO_PARAMS, place particle atposthat swims inz-direction. The particle handle should be calledparticle.
particle = system.part.add(
pos=pos, mass=HYDRO_PARAMS['mass'], rotation=3*[False],
swimming={'f_swim': HYDRO_PARAMS['active_force'],
'mode': HYDRO_PARAMS['mode'],
'dipole_length': HYDRO_PARAMS['dipole_length']})
system.integrator.run(HYDRO_N_STEPS)
vels = np.squeeze(lbf[:, int(system.box_l[1]/2), :].velocity)
vel_abs = np.linalg.norm(vels, axis=2)
lb_shape = lbf.shape
xs, zs = np.meshgrid(np.linspace(0.5, box_l[0] - 0.5, num=lb_shape[0]),
np.linspace(0.5, box_l[2] - 0.5, num=lb_shape[2]))
fig_vels, ax_vels = plt.subplots(figsize=(10, 6))
im = plt.pcolormesh(vel_abs.T, cmap='YlOrRd')
vels = np.copy(vels)
ax_vels.streamplot(xs, zs, vels[:, :, 0].T, vels[:, :, 2].T, density=1.5)
circ = plt.Circle(particle.pos_folded[[0, 2]], 0.5, color='blue')
ax_vels.add_patch(circ)
ax_vels.set_aspect('equal')
plt.xlabel('x')
plt.ylabel('z')
cb = plt.colorbar(im, label=r'$|v_{\mathrm{fluid}}|$')
plt.show()
We can also export the particle and fluid data to .vtk format to display the results with a visualization software like ParaView.
lbf.write_vtk_velocity('./fluid.vtk')
system.part.writevtk('./particle.vtk')
Further reading¶
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