Conclusions

Chapter III: Reverse Osmotic Effects in Active Matter

3.5 Conclusions

concentration gradient as pointed out by Brady [7].

Indeed active bath particles do not even require any spatial variations in transport properties to generate a directed motion of an immersed body. All that matters is a broken symmetry of the osmotic pressure at the contact surface, which can be also achieved by an asymmetric body shape of the immersed object (e.g. chevron, chiral gear, etc.) in active matter systems. From discussions in this section it is clear that the (classical) diffusiophoresis is just a special case of ‘osmophoresis’ [9], in which a concentration gradient is a source of the broken symmetry to generate nonzero total osmotic force on a phoretic body, from Brady’s new perspective. Thus, the reverse diffusiophoresis in an active bath can be considered as an immediate consequence of the reverse osmotic effect in active matter systems.

in biological systems (e.g. how the cell maintains its osmotic balance with its surroundings) may need to be reexamined.

It is important to be mindful that our intuition and knowledge obtained from passive or equilibrium systems can lead to wrong conclusions when dealing with active systems that are naturally driven away from the equilibrium. We considered diffusiophoresis in an active bath as an example of counterintuitive phenomena in active systems. A phoretic body in an active bath can move from regions of low concentration to high even though the phoretic and bath particles undergo purely repulsive interactions, which directly contradicts well-known results for diffusiophoresis in a passive bath. A phoretic motion is fundamentally powered by the total osmotic force on an immersed object [7] and therefore the unique reverse osmotic behavior of an active suspension generates the reverse diffusiophoresis when the swim speed of active bath particles spatially varies. From BD simulations the reverse diffusio- phoretic speed is shown to be linearly proportional to the gradient of run length of active bath particles as predicted by Brady [7].

In the following Chapter we discuss another example that is seemingly confounding from the equilibrium viewpoint: motility-induced phase separation. Instead of using equilibrium theories that are not guaranteed to be fruitful when describing active matter systems, we return to first principles — simple mechanics — and solve the problem that has intrigued and puzzled the active matter community since its beginning.

References

[1] W. Yan and J. F. Brady, “The force on a boundary in active matter,”J. Fluid Mech., vol. 785, R1.1–R1.11, Dec. 2015.

[2] A. K. Omar, Z.-G. Wang, and J. F. Brady, “Microscopic origins of the swim pressure and the anomalous surface tension of active matter,” Phys. Rev. E, vol. 101, no. 1, p. 012 604, 2020.

[3] S. C. Takatori and J. F. Brady, “Towards a thermodynamics of active matter,”

Phys. Rev. E - Stat. Nonlinear, Soft Matter Phys., vol. 91, no. 3, p. 32 117, 2015.

[4] W. Yan and J. F. Brady, “The swim force as a body force,”Soft Matter, vol. 11, no. 31, pp. 6235–6244, Jul. 2015.

[5] M. Cheryan,Ultrafiltration and Microfiltration Handbook. CRC press, 1998.

[6] Z. Peng, T. Zhou, and J. F. Brady, “Activity-induced propulsion of a vesicle,”

arXiv preprint arXiv:2112.05904, 2021.

[7] J. F. Brady, “Phoretic motion in active matter,”Journal of Fluid Mechanics, vol. 922, 2021.

[8] ——, “Particle motion driven by solute gradients with application to au- tonomous motion: Continuum and colloidal perspectives,” Journal of Fluid Mechanics, vol. 667, pp. 216–259, 2011.

[9] U. M. Córdova-Figueroa and J. F. Brady, “Osmotic propulsion: The osmotic motor,”Physical review letters, vol. 100, no. 15, p. 158 303, 2008.

[10] A.-Y. Jee, Y.-K. Cho, S. Granick, and T. Tlusty, “Catalytic enzymes are active matter,” Proceedings of the National Academy of Sciences, vol. 115, no. 46, E10812–E10821, 2018.

C h a p t e r 4

MECHANICAL THEORY OF PHASE COEXISTENCE

Liquid-gas phase separation is a well-understood topic in thermodynamics [1, 2].

From statistical thermodynamics, it can be shown that a liquid-gas phase separation requires attractive interactions as an essential ingredient [3], which is usually accepted as common knowledge. However, this is not true in active matter systems, which are inherently out of equilibrium due to the self-propelling motion of active constituents; purely repulsive active colloids can undergo a liquid-gas phase separation: the motility-induced phase separation (MIPS) [4–7]. While it seems confounding, the phenomenological description [5] is surprisingly simple: a posi- tive feedback between the repulsive collisions that reduces the local swim speed [8, 9] and slow swim speed that increases the number density [10–13].

Due to the nonequilibrium nature, the classical and statistical thermodynamic framework cannot be directly applied to analyze MIPS, and there have been a number of attempts to generalize thermodynamic frameworks to active systems for the description of MIPS [14–25]. However, debate continues about the correct definition of the (nonequilibrium) chemical potential and free energy of active matter systems, which are core thermodynamic properties required for a description of phase be- haviors. In this Chapter, we propose an alternative perspective to approach MIPS:

instead of generalizing the thermodynamic framework, we develop a mechanical theory of phase coexistence that can be generally applicable to equilibrium and nonequilibrium systems. In the following sections, we first revisit the thermodynamic descriptions of phase equilibrium and introduce an equivalent mechanical description of phase equilibrium. Unlike the thermodynamic framework, which only applies to equilibrium systems, a mechanical theory can be readily applied to the nonequilibrium phase coexistence. Using the mechanical approach, we reveal phase coexistence conditions and interface structure of MIPS.

4.1 Phase Equilibrium Conditions

Thermodynamics provides a powerful framework to study systems at equilibrium.

At the heart of thermodynamics is the thermodynamic extremum principles which is rooted in the second law of thermodynamics. The extremum principles state that the system possesses a minimum free energy at equilibrium. In some cases, the

minimum free energy state is achieved by having multiple phases in a system: phase equilibrium.

At phase equilibrium, the temperature𝑇, pressure 𝑝, and chemical potential 𝜇 of each species are uniform in all phases — the thermal, mechanical, and chemical equilibria. To see this, let us consider an isolated 𝐶-component system with 𝑁_𝑖 particles of species𝑖 (∈ {1,2, . . . , 𝐶}), volume𝑉, and energy of the system 𝐸 and suppose the system has two phases 𝛼 and 𝛽 at contact. We denote variables for each phase with superscripts e.g. 𝑉^𝛼 is the volume occupied by𝛼phase. Since the interface of the two phases is diathermal, movable, and permeable to the chemical species, the energy, volume, and number of particles of each phase are unconstrained.

Using the additivity of the entropy𝑆 =𝑆^𝛼+𝑆^𝛽and the fundamental thermodynamic relation, the variation in the system entropy to the first orders is given by

𝛿 𝑆 = 1

𝑇^𝛼

− 1 𝑇^𝛽

𝛿 𝐸^𝛼+

𝑝^𝛼 𝑇^𝛼

− 𝑝^𝛽 𝑇^𝛽

𝛿𝑉^𝛼−

𝐶

∑︁

𝑖=1

𝜇^𝛼

𝑖

𝑇^𝛼

− 𝜇

𝛽 𝑖

𝑇^𝛽

! 𝛿 𝑁^𝛼

𝑖 . (4.1)

From the second law of thermodynamics, the entropy of an isolated system reaches a maximum at equilibrium and thus the first differential of the system entropy 𝑆 with respect to any macrostate variable𝑋vanishes at an equilibrium state: 𝛿 𝑆/𝛿 𝑋 = 0. Consequently 𝛿 𝑆 = 0 for any small variations in unconstrained variables at equilibrium and therefore from eq. (4.1)

𝑇^𝛼 =𝑇^𝛽, 𝑝^𝛼 = 𝑝^𝛽, 𝜇^𝛼

𝑖 = 𝜇

𝛽

𝑖 ∀𝑖 ∈ {1,2, . . . , 𝐶}. (4.2) When more than two phases are present, the phase equilibrium condition (4.4) is satisfied by every pair of phases by the zeroth law of thermodynamics. Hence,𝑇, 𝑝, and 𝜇 are the same in all coexisting phases at equilibrium. We note that the Gibbs phase rule is an immediate consequence of a degree of freedom analysis on the phase equilibrium condition.

One can also find thermodynamic stability conditions by considering the second order variation of the entropy. For example, we consider variations in the energy:

𝛿²𝑆= 1 2

𝜕

𝜕 𝐸^𝛼 1

𝑇^𝛼

𝑉^𝛼, 𝑁^𝛼

+ 𝜕

𝜕 𝐸^𝛽 1

𝑇^𝛽

𝑉^𝛽, 𝑁^𝛽

(𝛿 𝐸^𝛼)². (4.3) Now we let the two phases to be identical — the system is homogeneous — and treat the interface as an imaginary surface dividing the system into two regions 𝛼

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<latexit sha1_base64="r4SZFCO+vFwtmUDoHpAn+4XjGww=">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</latexit>

Thermodyanmically

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unstable

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 volume𝑣 at a fixed temperature𝑇. The points at contact with the common tangent line (blue) correspond to the coexisting phases. The slope and 𝑦-intercept of the common tangent represents the the coexistence pressure (𝜕 𝐹/𝜕 𝑣)𝑇 = −𝑝 and the coexistence chemical potential 𝐹− 𝑣(𝜕 𝐹/𝜕 𝑣)𝑇 = 𝜇. The red region of the curve indicates unstable region from the stability condition (4.5). (b) A schematic description of the Maxwell construction on an isotherm in the𝑝−𝑣diagram. A horizontal line that cuts equal areas from the isotherm above (shaded blue) and below (shaded red) gives the coexistence pressure. The intersections of the horizontal line and isotherm correspond to the coexisting phases. The red region of the curve indicates unstable region from the stability condition (4.5).

and 𝛽. By the second law, the entropy of the system holds its maximum value at equilibrium. Thus,𝛿²𝑆 ≤0 and from eq. (4.3) we should have

𝑐_𝑉 = 1 𝑁

𝜕 𝐸

𝜕𝑇

𝑉 , 𝑁

≥0, (4.4)

where 𝑐^𝛼

𝑉 is the molar heat capacity at constant volume and𝑇 = 𝑇_𝛼 = 𝑇_𝛽 from the equilibrium condition (4.4). Using the thermodynamic extremum principles, similar variational analyses can be performed on systems with different conditions.

One of the most useful stability conditions is obtained by the Helmholtz energy minimization principle for the system at fixed𝑁 , 𝑉 ,and𝑇:

𝜅_𝑇 =−1 𝑉

𝜕𝑉

𝜕 𝑝

𝑇 , 𝑁

≥ 0, (4.5)

where 𝜅_𝑇 is the isothermal impressibility and 𝑝 = 𝑝_𝛼 = 𝑝_𝛽 from the equilibrium condition (4.4).

A phase diagram can be constructed by directly finding states that satisfy the phase equilibrium condition. When a pressure-explicit equation of state that is valid for multiple phases (e.g. van der Waals equations state for liquid and gas) is known for a single component system, alternative graphical representations of the equilibrium condition provide more convenient and instructive methods to obtain a phase diagram: the double-tangent and Maxwell (equal-area) constructions.

In the double-tangent construction, the molar Helmholtz free energy 𝐹 curve is drawn as a function of the molar volume𝑣at a constant temperature𝑇 as shown in Fig. 4.1(a). A tangent line of the free energy curve has the slope (𝜕 𝐹/𝜕 𝑣)𝑇 = −𝑝 and𝑦-intercept𝐹−𝑣(𝜕 𝐹/𝜕 𝑣)𝑇 =𝜇. Therefore if multiple points, say(𝑣^𝛼, 𝐹(𝑣^𝛼, 𝑇)) and(𝑣^𝛽, 𝐹(𝑣^𝛽, 𝑇)), on the curve share a common tangent, the corresponding states (𝑣^𝛼, 𝑇)and(𝑣^𝛽, 𝑇)satisfy the phase equilibrium condition and are called the binodal.

When the overall molar volume of the system 𝑣 ∈ (𝑣^𝛼, 𝑣^𝛽), the total Helmholtz free energy is minimized by having the two separated phases with (𝑣^𝛼, 𝑇) and (𝑣^𝛽, 𝑇): the Helmholtz free energy minimization principle for fixed 𝑁 , 𝑉, and𝑇. The phase separation always occurs when the free energy curve is convex upward, i.e. (𝜕²𝐹/𝜕 𝑣²)𝑇 = −(𝜕 𝑝/𝜕 𝑣)𝑇 < 0 — the red segment in Fig. 4.1(a) — to meet the mechanical stability condition (4.5). The points at the onset the instability, i.e. (𝜕²𝐹/𝜕 𝑣²)𝑇 = 0 are called the spinodal. The states between the binodal and spinodal is called metastable. In the metastable region, the system is allowed to exist as a homogeneous state without a phase separation since the stability condition (4.5) is satisfied. However, a phase separation would occur whenever possible in order to minimize the total Helmholtz free energy of the system.

The Maxwell construction is obtained by considering the first derivatives with respect to the molar volume𝑣in the double-tangent construction. We now consider the pressure 𝑝 = −(𝜕 𝐹/𝜕 𝑣)𝑇 instead of 𝐹 and draw a 𝑝−𝑣 isotherm as shown in Fig. 4.1(b). Then one seeks the coexistence pressure 𝑝 = 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇) such that

𝑝(𝑣^𝛼, 𝑇) = 𝑝(𝑣^𝛽, 𝑇) = 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇),

∫ ^𝑣^𝛽

𝑣^𝛼

[𝑝(𝑣 , 𝑇) − 𝑝^{𝑐𝑜 𝑒𝑥}(𝑇)] 𝑑 𝑣 =0, (4.6) then the intersections of the horizontal line and the isotherm gives the coexisting states. The first condition in (4.6) is trivially required to have the same pressure in the two states (𝑣^𝛼, 𝑇) and (𝑣^𝛽, 𝑇). The integral in (4.6) is required for the same chemical potential in the coexsiting phases. To see this, we integrate by parts to

Dalam dokumen Effect and Motility-Induced Phase Separation (Halaman 67-70)

Chapter III: Reverse Osmotic Effects in Active Matter

3.5 Conclusions

MECHANICAL THEORY OF PHASE COEXISTENCE

v

v

p

v a

v b

v a

v b

(a) (b)

p

F

v ^a

v ^b

v ^a

v ^b