Self-motion, which is the defining feature of active matter, is ubiquitous in living systems, and this is one of the reasons why research interest in active matter systems has increased significantly over the past decade. One of the main questions in the field of active matter research is about the consequences of self-motion.
Active Brownian Particles
The behavior of an active suspension is governed by the Péclet number𝑃𝑒=𝑈0ℓ0/𝐷𝑇 = (ℓ0/𝛿)2, which serves as a measure of the activity of ABPs. The evolution of the probability density in the phase space (𝒙,𝒒) is governed by Smoluchowski.
Mechanics of Active Suspensions
The usefulness of swimming pressure can be clearly revealed by considering the concept of equivalent pressure. When calculating the pressure exerted by the fluid on a wall, the "true" fluid pressure 𝑝 must be used, not the equivalent pressure.
Thesis Outline
As the activity 𝑃𝑒 decreases, the difference between the number densities in the two regions disappears. In both cases, the direction of the phoretic movement aligns with the total osmotic force 𝑭𝑜 𝑠𝑚 𝑜.
Active Matter with Spatially Varying Transport Properties
Introduction
We also present an interesting transient behavior of the Green's function of the Smoluchowski equation for active particles with a sudden change in activity. Since isothermal suspensions are considered, the variation of 𝐷𝑇 is equivalent to the spatially varying Stokes drag coefficient 𝜁, which depends on the suspended fluid viscosity and hydrodynamics.
Figure 2.12: (Left) Snapshots from BD simulations of purely active particles re- re-leased at 𝑡 = 0 from the center (white dashed line) where the reorientation time or swim speed abruptly changes
Conclusions
When 𝜏𝑅,1=𝜏𝑅,2regardless of the swimming speed in the two regions, the landmarks of the rescaled probability density 𝐴= 𝑃𝑈 are symmetric and also fall on a single curve on the rescaled domain 𝒙 =𝒙/ℓ. When the reorientation time during a suspension is uniform, the Smoluchowski equation can be rescaled so that it has no swimming speed and the transient distributions fall on a single symmetrical master curve regardless of the swimming speeds in the two regions.
Appendix: Brownian Dynamics Simulations of Active Brownian
The total pressure 𝑃 is the sum of the local osmotic pressure Π𝑜 𝑠𝑚 𝑜 = 𝑛 𝑘𝐵𝑇 exerted by particles and the fluid pressure 𝑝𝑓. The equilibrium condition for the system is given by the minimization of the large potential𝑊.
Reverse Osmotic Effects in Active Matter
Particle Phase Pressure on Walls
Since 𝑛𝑏𝑢𝑙 𝑘𝑈 is nearly constant for highly active particles, as shown in the previous chapter (see Figure 2.3), the pressure exerted by particles on the walls Π𝑤 ≈ Π𝑠𝑤𝑖 𝑚 ∼ 𝑛𝑏𝑢𝑙 𝑘𝜁 𝐷 𝑠𝑤𝑖 𝑚 ∼ 𝜁(𝑛𝑏𝑢𝑙 𝑘𝑈)ℓ will otherwise are in the two regions if the run lengths are differentℓ1 ≠ℓ2. In both cases, the ratio of the pressure exerted by particles on the walls is ℓ1/ℓ2 when the activity is high (𝑃𝑒≫ 1).
Mechanical Description of Active Matter with Spatially Varying Ac-
The drop in fluid pressure at the wall is exactly equal to the buoyant pressure exerted by the active particles on the boundary. c 937CM7dmw3FoHSlcrfqekHD2dKj2Yfzz15Ov/s+cLii2MV9RLGGywSUdJ0qeIiCHlDB1rwZpxwKl3BT9xuNWs/6fNEBVH4Uw9i3pLUD4N2MV9RLGGywSUdJ0qeIiCHlDB1rwZpxwKl3BT9xuNWs/6fNEBVH4Uw9i3pLUD 4N2KlXhOUd8bmkbmkbmqbmqbmqbmqbmqbmqbmqbmqvbmqbmqhvn z53gR60keaiaoUqd2JdYtQxMdMMGHc05P8ZiyLvX5aRqGVHLVMjDXofU+zXhWO0rSX6gtyI5XGCqVGkg37Smp7qjJtix5b5srb41sshB72oSJII x1sshB72oSJIIx1sshB72oSJIIx1ssb46ios2000sssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss. g4UyLQRpQlgTpK1msQxPKdLqKc07IL1kkJQ0942g+PLVbxpFudGUcNxJeNqa1Yi3b1spwONlZ3d/bOIHOKoZ3SmjfHytZcQQNfcGzvglEUODxthOEIUcMcLOb5VPEZW9MxthOEIW9McL0VeV9McL00000009 Pt6nBNR 603wmZNe6OWNMCkPIfc+U2CUVEzJvuzamO5ap6rYuE61K2jG+AGegO8gd4Eb6JrYBzDbYKb6D3wHnobvI2emLd7Bb5Cj1aheHsX7KLA7BKLx9WGx7KLXHxLxWLxWLxWLxWLxWLxWLxWLxWLxWLxWLxWLxWLx xWLxWLXWLxWLxWLxWLxWG 8CUa2EaqwJf5TkGIDZPj1MF1dAJO0KOPofgKJFii4UM1HvoIfITeB++jD8GH6GvwNXoHvIPeBe8WLwZWaB/so2NwjLbBNnoLvIW+AF+gq +DiQ6Qi7uQLCWExcTwK1+PHIJLcz/MQFgtd7Ej91o4cFB UH4xVbVKaHE+4B42THNE1AWXr/2pO37d3g+GPZ/lr+dPh5eXUtv4lnyRvyjnwgNvlGwohPMv9 9FRe85Lcekpv/wP7EUY9.
Figure 3.2: A schematic of a suspension of ABPs with an abrupt change in activity confined between two parallel planar walls (see Fig
Reverse Osmotic Effect in Active Matter
For spherical active particles with radii 𝑎 and Stokes drag coefficient𝜁 =6𝜋𝜂𝑎, |Δ𝑝𝑓| = (3/4)𝜂 𝜙𝑈ℎ|Δℓ|/𝑎2, where 𝜙 is the average volume fraction of suspended particles and 𝜂 is the viscosity of the liquid. Assuming a cylindrical connecting pipe and laminar flow, the pressure drop across the pipe is |Δ𝑝𝑡| = 𝜂𝑄 𝑅𝑡, where 𝑅𝑡 = 8𝐿𝑡/(𝜋𝑟4 . 𝑡) is the resistance of the pipe and 𝐿𝑡and𝑟𝑡 are the length and radius of the pipe.
Reverse Diffusiophoresis in Active Bath
The results obtained from the BD simulations confirm Brady's prediction [7] of reverse diffusiophoresis that the total osmotic force and thus the phoretic velocity is linearly proportional to the swimming velocity gradient. We also confirm that the magnitude of the total osmotic force is linearly proportional to the swimming velocity gradient.
Figure 3.4: A schematic of reverse diffusiophoresis in a bath of active hard-spheres.
Conclusions
Assume that two phases 𝛼 and 𝛽 coexist in equilibrium in the absence of external potential. From the particle force balance (4.49), the difference in collision pressures between the two phases is balanced by the integral of the buoyancy force: ΔΠ𝑐.
Mechanical Theory of Phase Coexistence
Phase Equilibrium Conditions
According to the second law, the entropy of a system has its maximum value at equilibrium. When the total molar volume of the system is 𝑣 ∈ (𝑣𝛼, 𝑣𝛽), the total Helmholtz free energy is minimized by two separate phases with (𝑣𝛼, 𝑇) and (𝑣𝛽, 𝑇): Helmholtz free energy minimization principle for fixed 𝑁 , 𝑉 and 𝑇.
Figure 4.1: (a) A schematic description of the double-tangent construction on an isotherm in the diagram of the molar Helmholtz free energy 𝐹 versus the molar volume 𝑣 at a fixed temperature 𝑇
Microscopic Structure of Phase Interfaces
Thus, the simple model of free energy density functional 𝑓 = 𝑓0(𝜌) only captures the macroscopic equilibrium condition (4.1) of classical thermodynamics - the microscopic structure of interfaces cannot be described. Equation (4.25) clearly shows that the interfacial tension is the result of the square gradient term which penalizes non-homogeneity in the density due to interparticle interactions.
Mechanical Description of Phase Interfaces
The stress deviates from −𝑝0𝑰 — the stress when the fluid is homogeneous — due to inhomogeneities in density. By integrating the balance of mechanical forces (4.29), we have 4.30) the pressure is the same deep within the coexisting phases due to the lack of density gradients in the bulk: a condition of mechanical equilibrium.
Motility-Induced Phase Separation
4.2 (a) and 4.2 (b). AAAGZXicbdTdbtMwFABgb7AyBoMNEDdckLEhcTUaQMDltO6H/WndSrdKSzU5jptmteMsdru1NTRbiHpHbDXuka crlv1PTDx7OlB7NPp578nT+2fOFxRcnWnVjxutMCRU3XKq5CE JeN4ERvBHFnEpX8FO3U8naT3s81oEKf5p+xJuS+mHQChg1aao2+LhxDVxYcB0 4YJqvWZXY5MM6GxCZjgwzmnq3lEWYf6/CwNQyq5biYw16H1Ps14VkvF6S80FmTHKxIqte5LN+0pqWnrybYseW+bK2+Nn GQ9jFLJCtHlDFuNFJL4HlKyNf mRD8NKIuD9JUs1qYxZSZdxTkn5FdMSUlDL3EMH57ZzcSRrrpOHFcJLxvTWrGWbWtlOJzsrO/vnTiBySqGd0pozx8rWXEEDX3Bs74xTRFDCG8Zy XFT4 vbS7WpzQ0etN8JmQ7ujljTApLyA3MVNglFRTZLsz6qO5Sp5roKFG1C3ga6D6+hN8CZ6C7yFroJxDLcBbqD3wfvoHfAOemLe7jXTXvGh1BXge z1a3GJlr8BXaGALqQNf5jsFITZMjlMD19AxOEaPPobiK5BgiYYPNfHQx+Bj9AH 4AH0EPkIPwAP0LngXvQfeK14MrNE+2EdH4Ahtg230NngbfAH4Ahtg230NngbfAhtg230NngbfAH4AH00NngbfAH4Ah CIuFLnakdmtHDouKw/GKbSrTwwn3QOJkxzRNQFl6/9qTt +3d4OTTqv119fPRl+W19fwmniVvyDvygdjkG1kj P0iV1AkjPvlFfpM /M/9K86VXpdejrtNTec1Lcuspvf0PF3lGlw==. To supply the large required swimming force required at high activity, the domain of integration with non-zero 𝑚𝑧, i.e. interface width, should be increased by. lt;latexit sha1_base64="oyTLWekrbJicR5Y3SKn5SrtN1qs=">AAAGpnicbdTdTtswFADgwEbHuj/YLncTVpCQJqFmm7ZdMsrP+BsF1lGJVJ3juqmpnYTYLSs2SvEbHmf 7sBIIrXa3+m5l98HCu9Gj+cfnJ02fPXywsvvyh/ EFIWYP6wg+bDlFMcI81NNeCNYOQEekIdub0a+n42ZCFivvedz0OWEsS1+NdTolOUu2Ft7HP9YcjkF YTol2No824vVCprlXhMu8GVh5UjPyqtxfnqN3x6UAyT1NBlDq3qoFuRSTUnAoWl+2BYgGhfeKy8yT0iGSqYXFcFTxeZKkum T9MfpGVh5UjPyqtxfnqN3x6UAyT1NBlDq3qoFuRSTUnAoWl+2BYgGhfeKy8yT0iGSqFcFTxeZKkumYXT9MfpGp4UAyT1NBlDq4 2hfV+oqRvS3c+tiHvBQDOPZvfTHQhT+2baSbPDQ0a1GCcBoSFPHsmkPZK0TCf9Ltseu6K +lMTrRLZm8bnVirJe244vOuma5rJZsczlOJ6erO6pRrRxHxln fmnsfCyF6yacfX9lKedViguPC9dO1kvzPclDjDZLt6TJNs9EY4rMkgG0kCTMoLyF3cJCgR9ShK/8z6RK6W52pYuAl1m+gGuIHeAm+ht13Hb8Fu64Cb4 Ht8Hb8Fu2003 XTO2AHvQHeQI / BY3Qf3C/qGSRY0W3wEJ01t+jsFfgKDewiFXdlvlMQ4sD0OqfgU3QIDtHZy1C8BRIs0fCiRh30CfgEfQg+RB+Dj9HX4Gv0HngPvMQrNAUHF0Lx4 AxYtIRNDLGwlhceP4KVxPfga+ZG6eh7BodLEjp7d25KioOJqs2CEy+T jhHIjs9DNNElC2Mn1oBVqO4nJyLFvTh/Dd4Me7Nevj2vvvjD5X1PjMfMhvcyGy +l1dK3UqN0lk2dnclrXhm3rtLP/7K/YPM=.
Figure 4.2: (a) Generalized equal-area construction from the mechanical theory (eqs. (4.53) and (4.56)) in the (Π 𝑎 𝑐𝑡 = [Π 𝑐 + Π 𝑠𝑤𝑖 𝑚 ] , Π 𝑐 ) plane and (b) Maxwell construction for ℓ 0 / 𝐷 = 35 × 2 −1/6 in 2D
For example, the functional derivative of the mean density with respect to the internal chemical potential gives the variance or fluctuation of the microscopic density: 4.65). Equations (4.64) and (4.65) clearly show that the mean density is a functional of the internal chemical potential in the grand canonical formulation.
Appendix: Orientational Moment Equations of Interacting Active
We first derive the evolution equation of the number density field to account for mass transport. The first term corresponding to the self-drive refers to the definition of the microscopic density of polar order.
Appendix: Effects of Correlation between Collisional Stress and
Bertin, "Lack of an equation of state for the non-equilibrium chemical potential of gases of active particles in contact," The Journal of Chemical Physics, vol. Waals, "The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density," Journal of Statistical Physics, vol.
Figure 4.7: A schematic of Π 𝑎 𝑐𝑡 -Π 𝑐 plot with a van-der Waals-like loop. The corre- corre-lation between the collisional stress and orientation leads to ∫
MATERIALS AND METHODS The effects of temperature on the duration and survival of egg, larval and pupal stages, adult female weight and fecundity, and reproductive potential were
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![Figure 4.2: (a) Generalized equal-area construction from the mechanical theory (eqs. (4.53) and (4.56)) in the (Π 𝑎 𝑐𝑡 = [Π 𝑐 + Π 𝑠𝑤𝑖 𝑚 ] , Π 𝑐 ) plane and (b) Maxwell construction for ℓ 0 / 𝐷 = 35 × 2 −1/6 in 2D](https://thumb-ap.123doks.com/thumbv2/123dok/10401816.0/87.918.271.645.104.678/figure-generalized-construction-mechanical-𝑐𝑡-𝑠𝑤𝑖-maxwell-construction.webp)
