T HEORY OF M IXTURES

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T HEORY OF M IXTURES

COURSE LECTURE NOTES

J

OSEF

M

ÁLEK

, O

ND ˇREJ

S

OU ˇCEK Charles University

Prague 2020

WE THANKKARELTUMA^ˇ , MICHALHABERA, VOJT ˇECH PATO ˇCKA AND MARKDOSTALÍK FOR THEIR HELP AND REMARKS.

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1 Theory of interacting continua - introduction

What is a mixture? Oxford English dictionary: mixture: “A product of mixing, a complex unity or aggre- gate (material or immaterial) composed of various ingredient orconstituent partsmixed together”.

and goes on as follows:

“... mixed state or condition; co-existenceof different ingredients or different groups or classes of things mutually diffusedthrough each other.”

Goal of the course

• Basic understanding of (derivation of) models that might be capable of describing the following processes (examples of mixtures around us):

– Geophysics- Thermohaline circulation - (convection due to both temperature and concentration variations), Porous media flow - flow of water and transport of solubles through soils, rocks,..., Avalanches, Pollution spreading and transport through environment, Melting of ice and freezing of meltwater in glaciers (phase transitions), Mineral phase transitions in rocks, Generation and extraction of magma, Liquefaction (change of strength of porous water saturated solids under seismic forcing),..

– Astrophysics- Plasmas and gaseous mixtures in the stars

– Biology- Flow of tracers/fluids through biological tissues, Processes at cell membranes, Biochem- ical reactions, Remodulation of bones, Blood coagulation, Cloth formation and dissolution, Flow of blood plasma,...

– Chemistry- Chemical reactions, Chemical equilibrium, Chemical kinetics

– Material sciences- Multi-phase flow, Flow of aerosols, dyes, Deformation of composite materials – ...

• Understanding of the assumptions required for the derivation of the models and developing framework for their possible generalizations.

• To arrive at three-dimensional thermodynamically consistent material models of mixtures which in- clude temperature effects.

• To understand how the models of Fick, Darcy, Forchheimer, Biot, Brinkman, Allen-Cahn, Cahn-Hilliard and Stephan fit into the general framework of the mixture theory.

• To provide a sufficiently colorful “palette” of material models to cover the real-world processes.

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Some names & literature

• Fick (1855), On liquid diffusion.

• Darcy (1856), Les fontaines publiques de la ville de Dijon.

• Muskat (1937), The Flow of Homogeneous Fluids through Porous Media.

• Truesdell and Toupin (1960), The Classical Field Theories.

• Truesdell and Noll (1965), The non-linear field theories of mechanics.

• Atkin and Craine (1976), Continuum Theories of Mixtures: Applications

• Bowen (1976), Theory of mixtures.

• Truesdell (1984), Rational thermodynamics.

• Müller (1985), Thermodynamics

• Samohýl (1982), Racionální termodynamika chemicky reagujících smˇesí.

• de Groot and Mazur (1984), Non-equilibrium Thermodynamics.

• Rajagopal and Tao (1995), Mechanics of Mixtures.

• Drew and Passman (1998), Theory of Multicomponent Fluids.

• Pekaˇr and Samohýl (2014), The thermodynamics of linear fluids and fluid mixtures.

• Bothe and Dreyer (2015), Continuum thermodynamics of chemically reacting fluid mixtures.

Two main approaches within the theory of interacting continua

• Approach 1 - a concept based on the principle of equi-presence (all components of the mixture are assumed to co-exist simultaneously in each point of the continuum; the foundations of this approach have been laid by Truesdell et al. (e.g. Truesdell and Toupin, 1960) and is sometimes referred to as Continuum mixture theory

• Approach 2- a concept relying on postulating classical single-component balance laws together with interface conditions at intermediate (sufficiently fine spatial) scale, where components of the mixture are distinguishable (e.g. at the scale of the pore-space in case of porous media flow). This is followed by a suitable averaging procedure, which delivers a continuum-like mixture framework; one of the classical references to this approach is Drew and Passman (1998). This approach is often referred to as Multi-phase theory.

In this lecture, we shall mostly rely on the first approach, with the exception of Chapter 8, where foundations of the second approach will be presented.

2 Reminder of the concepts of classical equilibrium thermodynamics

In this section, we first address as a reminder the formal structure of classical equilibrium thermodynamics, following closely Evans’ Lecture notes entropy and PDEs. (Evans, 2017). The development of classical physical notions is here performed in a mathematically formal way, a more "physical" introduction to the topic can be found, for example in Callen (1960). In the next chapters, these concepts are used under so-called assumption of local equilibrium used to transfer the thermodynamic relation to the realm of continuum physics.

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2.1 Physical postulates and mathematical model for a macroscopic system in thermo- dynamic equilibrium

Let us nevertheless start with the physical postulates (c.f. Callen, 1960, p. 283-284):

A physical model for a system in thermal equilibrium

• There exist particular states (called equilibrium states) that, macroscopically are characterized com- pletely by the specification of internal energyEand set of extensive parametersX₁, . . .X_m

• There exists a function (calledentropy) of the extensive parameters, defined for all equilibrium states, with the following property: the value assumed by the extensive parameters in the absence of constraints, are those that maximize the entropy over the manifold of equilibrium states

• The entropy of a composite system is addition over the constituent subsystems (hence the entropy of each system is a homogeneous first order function of the extensive parameters. The entropy is continuous and differentiable and is monotonically increasing function of energy

• The entropy of any system vanishes in the state for whichT:=^∂_∂S^E =0

Let us translate these physical postulates into a set of mathematical postulates A mathematical model for a system in thermal equilibrium

Let us suppose we are given

• an open convex subsetΣofR^m⁺¹

• aC¹ function ˆS:Σ→Rsuch that (i) ˆSis concave

(ii) ^∂_∂E^S^ˆ >0

(iii) S is positively homogeneous of degree 1:

S(ˆ _λE,_λX₁, . . . ,X_m)=λS(E,ˆ X₁, . . . ,X_m) _λ>0 (2.1) We callΣthe state space andSthe entropy of the system

S^def= S(E,ˆ X₁, . . . ,X_m) Fundamental thermodynamic equation (2.2) depending on the internal energyEand other extensive quantities X₁, . . . ,X_m (examples of such quantities are for example the volumeV, number (or molar number) of particles of individual atoms or moleculesN_αor their massesMα, electric polarizationP, magnetizationM, etc.)

Owing to (ii), we can invert relation (2.2) and obtain aC¹function ˆEas a function

E=E(S,ˆ X₁, . . . ,X_m). (2.3)

We define:

• Thermodynamic temperature T

T=Tˆ=^∂ Eˆ

∂S (2.4)

• Generalized force (or pressure)

P_k=Pˆ_k= − ^∂ Eˆ

∂X_k (2.5)

Lemma 2.1. (i) The functionE is positively homogeneous of degree 1:ˆ

E(ˆ _λS,_λX₁, . . . ,_λX_m)=λE(S,ˆ X₁, . . . ,X_m) _λ>0. (2.6)

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(ii) The functionsT,ˆ Pˆ_k are positively homogeneous of degree 0:

Tˆ(_λS,_λX₁, . . . ,_λX_m)=Tˆ(S,X₁, . . . ,X_m) (2.7) Pˆ_k(_λS,_λX₁, . . . ,_λX_m)=Pˆ_k(S,X₁, . . . ,X_m) _λ>0 (2.8) These properties will leads to distinguishing so-calledextensive parameters:S,E(and alsoX_k), which- so-to-speak depend on how “big” the system is andintensive parametersT, P_k, which do not depend on the “size” of the system.

Proof. 1. Clearly, it must hold for allE,X₁, . . . ,X_m:

E=E( ˆˆ S(E,X₁, . . .X_m),X₁, . . . ,X_m). (2.9) Thus for all_λ>0:

λE=E( ˆˆ S(_λE,_λX₁, . . ._λX_m),_λX₁, . . . ,_λX_m)

=E(ˆ _λS(E,ˆ X₁, . . . ,X_m),_λX₁, . . . ,_λX_m) (2.10) due to (2.1)

2. Since ˆSisC¹, so is ˆE. Differentiate (2.6) with respect toS:

λ∂Eˆ

∂S(_λS,_λX₁, . . . ,_λX_m)

| {z }

T(λS,λX₁,...,λX_m)

=λ∂Eˆ

∂S(S,X₁, . . . ,X_m)

| {z }

T(S,X₁,...,X_m)

(2.11)

Lemma 2.2. It holds

∂Sˆ

∂E = 1

T, ^∂

Sˆ

∂X_k =P_k

T k=1, . . . ,m (2.12)

Proof. We definedT asT=^∂_∂^E_S^ˆ and the function ˆS(E,X₁, . . . ,X_m) as an inverse ofE=E(S,ˆ X₁, . . . ,X_m). Thus it holds

E=E( ˆˆ S(E,X₁, . . . ,X_m),X₁, . . . ,X_m) . (2.13) Therefore

1=dE dE=^∂

Eˆ

∂S

∂Sˆ

∂E=⇒T=^∂ Eˆ

∂S = Ã∂Sˆ

∂E

!−1

. (2.14)

Similarly, by taking a derivative with respect toX_k, we get 0= ^∂

Eˆ

∂S

|{z}

T

∂Sˆ

∂X_k+ ^∂ Eˆ

∂X_k

| {z }

−Pk

k=1, . . . ,m. (2.15)

The definitions (2.4) and (2.5) can be summarized by the so-calledGibbs’s formula dE=T dS−

m

X

k=1

P_kd X_k (2.16)

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2.2 Examples

The extensive parameters X₁, . . . ,X_m can represent various physical quantities. Let us discuss some proto- typical examples

1. Simple fluid. Possibly the simplest example of an equilibrium thermodynamic system is a homoge- neous simple fluid, for which the extensive parameters are:

• E- internal energy

• X₁=V - volume

• X₂=N - number of particles

consequently, the fundamental thermodynamic relation for such system readsS=S(E,Vˆ ,N), and the associated intensive parameters are

• T=^∂_∂^E_S^ˆ - thermodynamic temperature

• P₁=P= −^∂_∂_V^E^ˆ - pressure

• P₂= −µ= −_∂N^∂^E^ˆ - (molar) chemical potential

The Gibbs formula for homogeneous simple fluid thus reads dE=T dS−P dV+µdN .

2. Multicomponent simple fluid. If the fluid is composed of N constituents, we are in the following setting. The extensive parameters are now

• E- internal energy

• X₁=V - volume

• X₂, . . . ,X_N+1=N1, . . . ,NN - numbers of particles of the individual constituents

consequently, the fundamental thermodynamic relation for such system readsS=S(E,ˆ V,N1, . . . ,NN), and the associated intensive parameters are

• T=^∂_∂^E_S^ˆ - thermodynamic temperature

• P₁=P= −^∂_∂_V^E^ˆ - pressure

• P₂, . . .P_N₊₁:P_α+₁= −µ_α= −_∂N^∂^E^ˆ_α - chemical potentials_α=1, . . . ,N The Gibbs formula for such a multicomponent simple fluid thus reads

dE=T dS−P dV+

N

X

α=1

µ_αdNα.

3. Homogeneous single-component dielectric fluid in a homogeneous electrostatic field with intensityE. From the principle of virtual work, one can deduce that the change of total energy of the volume filled with the dielectric is

dE=T dS−P dV+µdN + Z

ΩE·dD.

whereDis the electric induction vector. Assuming a homogenous field, this relation can be written as dE=T dS−P dV+µdN +VE·dD.

For an isotropic linear dielectric body readsD=ε0E+P, wherePis the so-called polarization vector and ε0 is the permitivity of vacuum. Often one introduces a reduced energy ˜E:=E−V^|^E₈_π^|² which results

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is obtained by subtracting the energy of the electrostatic field (it cannot be however interpreted as intrinsic energy of the dielectric since the value ofEis affected by the presence of the dielectric).

The Gibbs relation in terms of ˜Ereads

dE˜=T dS−P dV+µdN +VE·dP

and, by introducing the extensive quantity total polarizationP :=PV, we can rewrite it as dE˜=T dS−P dV˜ +µdN +E·dP ,

with ˜P=P+^ε₂⁰|E|² +E·P, and consider fundamental relation ˜E=E(S,b˜ V,N,P).

2.3 Physical interpretation of the mathematical model

Let us spend some time with an attempt of deeper explanation of the mathematical assumptions laid in our model and their physical interpretations and consequences.

• Theassumption ^∂_∂_E^S^ˆ >0 impliesT>0, i.e. positivity of the thermodynamic temperature

• The1-homogeneity assumption ofSˆ is equivalent toadditivity of entropy over subsystems of an equilibrium system. Indeed, consider a bodyB in equilibrium, with internal energyE, entropy S and other extensive variablesX_k,k=1, . . . ,m. Let us consider a subregionB⁽¹⁾, which represents a λfraction ofB, and consequently, its extensive parameters take values: E⁽¹⁾=λE, X₁⁽¹⁾=λX₁. . .X_m⁽¹⁾= λX_m, and alsoS⁽¹⁾=λS, due to 1-homogeneity of ˆS, since

S⁽¹⁾=S(Eˆ ⁽¹⁾,X₁⁽¹⁾, . . . ,X_m⁽¹⁾)=S(ˆ _λE,_λX₁, . . . ,_λX_m)=λS(E,ˆ X₁, . . . ,X_m)=λS.

The complementary subregionB⁽²⁾(such thatB=B⁽¹⁾∪B⁽²⁾), has extensive parametersE⁽²⁾=(1−λ)E, X⁽²⁾₁ =(1−λ)X₁. . .X_m⁽²⁾=(1−λ)X_mand thusS⁽²⁾=(1−λ)Sby 1-homogeneity of ˆS. Consequently, we get S=S⁽¹⁾+S⁽²⁾. The argument can be reversed and by assuming additivity of entropy over subsystems of an equilibrium system, we would deduce 1-homogeneity of ˆS: Take division ofB intoN subsystems each of the same size, i.e. _N¹ fraction ofB. By the additivity assumption, it must hold

S=

N

X

i=1

S⁽ⁱ⁾=NSˆ µ1

NE, 1

NX₁, . . . , 1 NX_m

¶ ,

which must hold for allN∈N. Using the same argument for subsystems of the size ^M_N,M∈{1, . . . ,N}, we obtain thatqS(E,ˆ X₁, . . . ,X_m)=S(qE,ˆ qX₁, . . . ,qX_m) for allq∈Q,q>0. By the assumption of continuity of ˆSand by the density argument ofQinR, we obtain the desired result.

• On the other hand, we have proven the intensive parametersT,P₁, . . . ,P_m, to be 0-homogeneous. This implies that these parameters take the same value irrespective of the size of the subregion, i.e.

T=T⁽¹⁾=T⁽²⁾=. . . ., P_k=P⁽¹⁾_k =P⁽²⁾_k =. . . . k=1, . . . ,m.

• Concavity of S. Let us now consider two isolated bodies made out of the same substanceˆ B⁽¹⁾ and B⁽²⁾which do not form subregions of one equilibrium system, but each of them is in equilibrium. Let

S⁽¹⁾=S(Eˆ ⁽¹⁾,X₁¹, . . . ,X¹_m) S⁽²⁾=S(Eˆ ⁽²⁾,X₁², . . . ,X_m²)

The total entropy of the two systems in this configuration isS^(1)+(2)=S⁽¹⁾+S⁽²⁾. If we now combine the two bodies into one (e.g. by allowing transfer of all extensive parameters such as energy, mass,...) but without doing any work, neither allowing heat transfer from outside, we end up with a systemB⁽³⁾ with extensive parametersE⁽³⁾=E⁽¹⁾+E⁽²⁾, X⁽³⁾_k =X⁽¹⁾_k +X_k⁽²⁾,k=1, . . . ,m. Since irreversible processes may have taken place inside the systemB⁽³⁾before it equilibrated, we cannot assume more than

S⁽³⁾≥S⁽¹⁾+S⁽²⁾

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i.e. we assume that the final equilibrium entropy of the joint system is greater or equal than the sum of the two entropies of the originally isolated subsystems. So this means that

S⁽³⁾=S(Eˆ ⁽¹⁾+E⁽²⁾,X₁⁽¹⁾+X₁⁽²⁾, . . . ,X_m⁽¹⁾+X_m⁽²⁾)≥S⁽¹⁾+S⁽²⁾=S(Eˆ ⁽¹⁾,X₁⁽¹⁾, . . . ,X⁽¹⁾_m)+S(Eˆ ⁽²⁾,X⁽²⁾₁ , . . . ,X_m⁽²⁾) . This however implies that ˆSis concave, since taking any 0<λ<1:

Sˆ³

λE⁽¹⁾+(1−λ)E⁽²⁾,_λX₁⁽¹⁾+(1−λ)X₁⁽²⁾, . . . ,_λX_m⁽¹⁾+(1−λ)X⁽²⁾_m´

≥Sˆ³

λE⁽¹⁾,_λX₁⁽¹⁾, . . . ,_λX_m⁽¹⁾´ +Sˆ³

(1−λ)E⁽²⁾, (1−λ)X⁽²⁾₁ , . . . , (1−λ)X_m⁽²⁾´

=λSˆ³

E⁽¹⁾,X₁⁽¹⁾, . . . ,X_m⁽¹⁾´

+(1−λ) ˆS³

E⁽²⁾,X⁽²⁾₁ , . . . ,X⁽²⁾_m´ where in the last equality, we employed the 1-homogeneity of ˆS.

• Convexity ofEˆas a consequence of concavity of ˆS. Let us take anyS⁽¹⁾,X₁⁽¹⁾, . . .X⁽¹⁾_m andS⁽²⁾,X⁽²⁾₁ , . . .X⁽²⁾_m and any 0<λ<1. Define

E⁽¹⁾=E(Sˆ ⁽¹⁾,X₁⁽¹⁾, . . .X_m⁽¹⁾) E⁽²⁾=E(Sˆ ⁽²⁾,X₁⁽²⁾, . . .X_m⁽²⁾) and thus

S⁽¹⁾=S(Eˆ ⁽¹⁾,X₁⁽¹⁾, . . .X_m⁽¹⁾) S⁽²⁾=S(Eˆ ⁽²⁾,X₁⁽²⁾, . . .X⁽²⁾_m) . Since ˆSis concave, it must hold

Sˆ³

λE⁽¹⁾+(1−λ)E⁽²⁾,_λX₁⁽¹⁾+(1−λ)X₁⁽²⁾, . . . ,_λX⁽¹⁾_m +(1−λ)X⁽²⁾_m´

≥λS(Eˆ ⁽¹⁾,X₁⁽¹⁾, . . . ,X⁽¹⁾_m)+(1−λ) ˆS(E⁽²⁾,X₁⁽²⁾, . . . ,X_m⁽²⁾) . Since it holds

E=E( ˆˆ S(E,X₁, . . . ,X_m),X₁, . . . ,X_m) and since ^∂_∂^E_S^ˆ =T≥0, we obtain

λE⁽¹⁾+(1−λ)E⁽²⁾=Eˆ³

S(ˆ _λE⁽¹⁾+(1−λ)E⁽²⁾, . . . ,_λX⁽¹⁾_k +(1−λ)X⁽²⁾_k , . . . ), . . . ,_λX_k⁽¹⁾+(1−λ)X_k⁽²⁾, . . .´

≥Eˆ³

λS(Eˆ ⁽¹⁾, . . . ,X⁽¹⁾_k , . . . )+(1−λ) ˆS(E⁽²⁾, . . . ,X_k⁽²⁾, . . . ), . . . ,_λX⁽¹⁾_k +(1−λ)X⁽²⁾_k , . . .´

=E(ˆ _λS⁽¹⁾+(1−λ)S⁽²⁾, . . . ,_λX_k⁽¹⁾+(1−λ)X_k⁽²⁾, . . . ).

So we obtained

λE(Sˆ ⁽¹⁾,X₁⁽¹⁾, . . . ,X_m⁽¹⁾)+(1−λ) ˆE(S⁽²⁾,X₁⁽²⁾, . . . ,X⁽²⁾_m)≥E(ˆ _λS⁽¹⁾+(1−λ)S⁽²⁾, . . . ,_λX⁽¹⁾_k +(1−λ)X⁽²⁾_k , . . . ) which implies that ˆEis convex.

• Entropy maximization and energy minimizationfor thermally isolated systems.

– Entropy maximization principle - The equilibrium value of any unconstrained internal pa- rameter is such as to maximize the entropy of the system for the given value of total internal energy.

– Energy minimization principle- The equilibrium value of any unconstrained internal param- eter is such as to minimize the energy for the given value of the total entropy.

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Figure 1: Entropy function ˆS(E, . . . ,X_k, . . . ) and visualization of the entropy maximization (left) end energy minimization (right) principle. The unconstrained variableX_kattains equilibrium valueX^∗_k, such that either entropy is maximal provided that the energy is fixed at valueE^∗ (left), or such that the energy is minimal provided that the entropy is fixed at valueS^∗.

2.4 Other thermodynamic potentials - Legendre transform

Assume that H:Rⁿ→(−∞,+∞] is a convex, lower semicontinuous function, which is proper (i.e. not identi- cally equal to infinity).

Definition 1. The Legendre transform ofH(p) denotedH^∗(q) is defined as H^∗(q)=sup

p∈Rⁿ(p·q−H(p)) (2.17)

One can show, that alsoH^∗(q) is convex, lower semicontinuous and proper. Furthermore,

(H^∗)^∗=H, (2.18)

i.e.His a Legendre transform ofH^∗, that’s why we call the pairH,H^∗ as dual convex functions.

Provided that on top of the assumptions before,His moreover also aC²and strictly convex, then for each q∈Rⁿthere exists a unique pointpwhere (p·q−H(p)) is maximal, namely the unique pointp, where

q=D_pH(p) , providing relationq=q(p)ˆ (2.19) which can be inverted to give

p=p(q).ˆ (2.20)

The Legendre transform then reads

H^∗(q)=p(q)ˆ ·q−H( ˆp(q)) (2.21) As a consequence, one obtains also

D_qH^∗(q)=p+¡

q−D_pH(p)¢

D_qpˆ=p, (2.22)

so to summarize

q=D_pH(p) p=D_qH^∗(q) (2.23)

So far, we have seenS=S(E,ˆ X₁, . . . ,X_m) andE=E(S,ˆ X₁, . . . ,X_m). Now Legendre transform is used to define otherthermodynamic potentials

• The Helmholtz free energyF is

F(T,V, . . . )^def= inf

S ( ˆE(S,V, . . . )−T S) (2.24)

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• The enthalpyHis

H(S,P, . . . )^def=inf

V ( ˆE(S,V, . . . )+PV) (2.25)

• The Gibbs free energyGis

G(T,P, . . . )^def= inf

S,V( Ê(S,V, . . . )−T S+PV) (2.26) Assuming Êis strictly convex andC²and that the infima are attained at a unique point in the domain of Ê, we can rewrite the potentials as

• The Helmholtz free energy

F(T,V, . . . )=E−T S, where S=S(T,ˆ V, . . . ) solves T=^∂

E(S,ˆ V, . . . )

∂S (2.27)

• The enthalpy

H(S,P, . . . )=E+PV, where V=V(S,ˆ P, . . . ) solves P= −^∂

E(S,Vˆ , . . . )

∂V (2.28)

• The Gibbs free energy

G(T,P, . . . )=E−T S+PV, where S=S(Tˆ ,P, . . . ) solves T=^∂

E(S,Vˆ , . . . )

∂S , P= −^∂

E(S,Vˆ , . . . )

∂V (2.29) Based on the definitions and the assumption ˆEis strictly convex andC², we can prove the following

Lemma 2.3. 1. E(S,ˆ V, . . . )is locally strictly convex in(S,V)

2. Fˆ(T,V, . . . )is locally strictly concave in T, locally strictly convex in V . 3. H(S,ˆ P, . . . )is locally strictly concave in P, locally strictly convex in S.

4. G(T,ˆ P, . . . )is locally strictly concave in(T,P).

Proof. First, the statement (1) is already assumed hence it is true. To prove (2), let us recall the definition of F

Fˆ(T,V, . . . )=E( ˆˆ S(T,V, . . . ),V, . . . )−TS(Tˆ ,V, . . . ) , with

T=^∂

E( ˆˆ S(T,V, . . . ),V, . . . )

∂S (2.30)

This implies

∂Fˆ

∂T =^∂ Eˆ

∂S

∂Sˆ

∂T −Sˆ−T^∂ Sˆ

∂T = −S(T,ˆ V, . . . )

∂Fˆ

∂V =^∂ Eˆ

∂S

∂Sˆ

∂V +^∂ Eˆ

∂V −T^∂ Sˆ

∂V =^∂ Eˆ

∂V( ˆS(T,V, . . . ),V, . . . )= −P(T,ˆ V, . . . ) which implies

∂²Fˆ

∂T²= −^∂

S(Tˆ ,V, . . . )

∂T (2.31)

∂²Fˆ

∂V²= −^∂

P(T,ˆ V, . . . )

∂V = ^∂

2Eˆ

∂S_∂V

∂S(Tˆ ,V, . . . )

∂V +^∂

2Eˆ

∂V² (2.32)

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Differentiating (2.30) with respect to (T,V), we obtain 1=^∂

2Eˆ

∂S²

∂S(Tˆ ,V, . . . )

∂T 0=^∂

2Eˆ

∂S²

∂S(Tˆ ,V, . . . )

∂V + ^∂

2Eˆ

∂V_∂S Thus (2.31) and (2.32) imply

∂²F

∂T² = − µ_∂2Eˆ

∂S²

¶−1

∂²F

∂V² =^∂

2Eˆ

∂V²− µ _∂2Eˆ

∂S_∂V

¶²µ_∂2Eˆ

∂S²

¶−1

and since we assumed that ˆEis strictly convex in (S,V, . . . ), it holds

∂²Eˆ

∂S² >0, ^∂

2Eˆ

∂V² >0

µ_∂2Eˆ

∂S²

¶ µ_∂2Eˆ

∂V²

¶

− µ _∂2Eˆ

∂S_∂V

¶²

>0 , and thus

∂²Fˆ

∂T²<0 , ^∂

2Fˆ

∂V²>0 . (2.33)

The remaining two statements are proved analogously.

Exercise1. Prove the statements (3) and (4).

3 Reminder of some basic concepts of classical continuum thermody- namics and mechanics of single continuum

3.1 Basic cornerstones of continuum thermodynamics

• A priori homogenization- continuum mechanics introduces the notion ofa material pointas a point- wise representative of some sufficiently small/sufficiently large control volume of real material; material properties assigned to such material point are averages (volume/time/stochastic) of the properties of the real material contained in the control volume.

X referential conf.

x=χ(X,t)

current conf.

B

κ0(B)

κt(B)

Figure 2: Essential kinematical concept of continuum mechanics - notion of motion_χof an abstract bodyB, viewed as a mapping from some reference configuration_κ₀(B)⊂R³ to the current configuration_κ_t(B)⊂R³ at given timet.

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• Kinematics

Motion - mapping

χ(·,t) :_κ₀(B)→κt(B) (3.1) R³×R→R³:x=χ(X,t) x^k=χ^k(X^K,t) , (3.2) which is assumed to be sufficiently smooth and invertible.

– Spatial gradient

GradΦ(X,t) = ^∂Φ(X,t)

∂X (3.3)

grad_φ(x,t) = ^∂φ

∂x (3.4)

– Velocity

V(X,t) = ^∂χ

∂t

¯

¯_X

in Lagrangean description (3.5)

v(x,t) = V(_χ⁻¹(x,t),t) in Eulerian description (3.6) – Material time derivative

D

D tΦ(X,t) = ^∂Φ(X,t)

∂t

¯

¯_X

(3.7) D

D t^φ(x,t) = ^∂φ(_χ(X,t),t)

∂t

¯

¯_X= ^∂φ(x,t)

∂t

¯

¯_x+v(x,t)·grad_x_φ(x,t) (3.8) – Deformation gradient

F(X,t)=^∂χ(X,t)

∂X (F)^k_K(X,t)=^∂χ

k(X,t)

∂X^K (3.9)

– Green deformation tensor

C(X,t) = F^TF (3.10)

(C)_{I J} = (F)ⁱ_I(F)^j_J_δ_{i j} (3.11) (3.12) – Cauchy deformation tensor

c(x,t) = F^−TF⁻¹ (3.13)

(c)_{i j} = (F⁻¹)^I_i(F⁻¹)^J_j_δ_{I J} (3.14) – Finger deformation tensor

B(X,t) = FF^T (3.15)

(B)^{i j} = (F)ⁱ_I(F)^j_J_δ^{I J} (3.16) – Piola deformation tensor

b(x,t) = F⁻¹F⁻^T (3.17)

(b)^{I J} = (F⁻¹)^I_i(F)^J_j_δ^{i j} (3.18) – Velocity gradient

L(x,t) = gradv (3.19)

(L)ⁱ_j = ^∂vⁱ

∂x^j (3.20)

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Exercise2. Show that ˙F=LF. Exercise3. Show that ˙B=LB+BL^T.

• Volume and mass measures, constraints

Consider a mixture bodyΩand letP ⊂Ωopen subset, then we define

∀Pt“nice” M(P) . . . mass contained inP V(P) . . . volume ofP

Constraint: ∀Pt: 0= _dt^dV(Pt) (using the substitution theorem and identity ˙

det(F)=divvdetF), we obtain the incompressibility constraint 0=R

Ptdivv=0, localization gives divv=0

• Balance equations (in Eulerian description) – Balance of mass

∂ρ

∂t +div(_ρv)=0 , (3.21)

where_ρis the density,vis the velocity.

– Balance of linear momentum

∂(_ρv)

∂t +div(_ρv⊗v)=divT+ρb, (3.22) whereTis the Cauchy stress,bis the intensity of the body forces.

– Balance of angular momentum (for non-polar continuum)

T=T^T. (3.23)

– Balance of total energy

∂

∂t µ

ρ µ

e+1 2|v|²

¶¶

+div µ

ρ µ

e+1 2|v|²

¶ v

¶

=div (Tv−q)+ρb·v+ρr, (3.24) whereeis the specific internal energy, andris the energy supply (e.g. radiation).

In the framework of continuum thermodynamics, one has to consider also – Balance of entropy

∂(_ρη)

∂t +div¡ ρηv¢

+divq_η+ρr_η=ξ, where _ξ≥0 . (3.25) Here _ηis the specific entropy, q_η is the entropy flux, r_η is the specific entropy supply, and _ξis the entropy production, which, by the second law of thermodynamics must be non-negative,

Exercise4. Show that (3.21)-(3.24) allow to rewrite (3.24) in a more compact form (balance of internal energy)

ρe˙=T:D−divq+ρr. (3.26)

Assuming the body forcesband energy supplyrare given, the balance equations (3.21), (3.22), (3.24) represent system of of 5 evolutionary equations for unknowns_ρ,e,v=(v₁,v₂,v₃), T=(T11,T12, T13, T22,T23,T33),q=(q₁,q₂,q₃) (14 unknowns), and it is known that it is in general impossible to predict the evolution of theTand q from this system knowing the initial state and boundary conditions. It is necessary to provide closure relations - relations describing howTand qdepend on the mechanical and thermal state of the system.

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3.2 Local equilibrium thermodynamics

In this course we shall rarely leave the realm of so called local equilibrium thermodynamics - a thermodynamic theory in which each material point (representing sufficiently small subsystem of the whole system), is assumed to be in local thermodynamic equilibrium. By saying local, we mean, that this equilibrium only concerns the subsystem, but naturally, this material point behaves as an open subsystem, exchanging mass, energy, entropy etc., with the neighbouring points, with which equilibrium may not have been reached.

Based on this idea, we can exploit the notions and relations that have been derived for macroscopic systems in thermodynamic equilibrium and simply apply them on some small representative volume of our system. We will then deduce some local relations, and postulate them to hold point-wise in our continuum model. In the following application we will be exclusively interested in the description of fluid mixtures in the absence of electric and magnetic fields. We shall often tacitly assume this without further emphasizing this point explicitly.

3.2.1 Entropic representation (for fluid mixtures)

So consider this volume element dΩ, assumed to be in thermodynamic equilibrium, for which we apply the formal equilibrium structure, briefly reminded in Section 2. So again, we postulate existence of a function S called “entropy”, which is a function of energy of the system, and it extensive variables - in most cases we will restrict ourselves to volume, and masses of individual components of the system:

S=S(E,ˆ V,Mα) , (3.27)

or more precisely

S(dΩ)=S(E(dˆ Ω),V(dΩ),Mα(dΩ)), (3.28) where E(dΩ), V(dΩ) andMα(dΩ) are the energy, volume and masses of component in the subdomaindΩ. The function ˆSwas postulated (see (2.1)) to be

• positive 1−homogeneous with respect to the extensive variables, meaning

S(ˆ _λE,_λV,_λMα)=λS(E,ˆ V,Mα) ∀λ>0 . (3.29)

• increasing function of energy

∂Sˆ

∂E>0 (3.30)

• ˆSis concave

From (3.27), we get by differentiating dS=^∂

Sˆ

∂EdE+^∂ Sˆ

∂VdV+

N

X

α=1

∂Sˆ

∂Mα

dM_α. (3.31)

In chapter 2, we defined the thermodynamic temperature_ϑ, thermodynamic pressurepand chemical poten- tials_µ_αby relations

1 ϑ=^∂

Sˆ

∂E , p ϑ=^∂

Sˆ

∂V , −^µ^α ϑ = ^∂

Sˆ

∂Mα , (3.32)

so we can rewrite (3.31) in the classical form

ϑdS=dE+pdV−

N

X

α=1

µ_αdMα . (3.33)

Differentiating (3.29) w.r.t. _λat_λ=1, we obtained the correspondingEuler relation S(E,ˆ V,Mα)=^∂

Sˆ

∂EE+^∂ Sˆ

∂VV+

N

X

α=1

∂Sˆ

∂MαMα, (3.34)

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or, equivalently, using the introduced definitions

ϑS=E+pV−

N

X

α=1

µ_αMα . (3.35)

Differentiating (3.35) and using (3.31), we obtained theGibbs-Duhem relation:

Sd_ϑ=V d p−

N

X

α=1

Mαd_µ_α . (3.36)

Let us now explore the consequences of 1-homogeneity of entropy:

• Let us first take _λ:= 1

M , withM:=PN α=1Mα: Sˆ

µ E M, V

M,Mα

M

¶

= 1

MS(E,ˆ V,Mα) . (3.37)

Recognizing in the arguments of the function on the l.h.s., the specific energy e, inverse of density of the mixture ¹_ρ and concentrations (mass fractions) c_α, defined as

e^def= E

M , 1 ρ

def= V

M , c_α^def= Mα

M ^α=1, . . . ,N, (3.38)

and observing that the function on the l.h.s. is a specific entropy_η, we can write this relation as 1

MS(E,ˆ V,Mα)=η^def=_ηˆ(e,1

ρ,c₁, . . . ,c_N) , (3.39) differentiating the last equality yields

d_η=^∂_ηˆ

∂ed e+^∂_ηˆ

∂¹_ρd µ1

ρ

¶ +

N

X

α=1

∂_ηˆ

∂c_αd c_α (3.40)

Dividing the Gibbs-Duhem relation (3.36) byM, we get ηd_ϑ=1

ρd p−

N

X

α=1

c_αd_µ_α . (3.41)

Dividing the Euler relation (3.35) byM, we obtain its local form ϑη=e+p

ρ−

N

X

α=1

µ_αc_α. (3.42)

Finally, differentiating (3.42) and subtracting (3.41), we recover the local version of (3.33):

ϑd_η=d e+pd µ1

ρ

¶

−

N

X

α=1

µ_αd c_α . (3.43)

Note that (3.43) implies the following local definitions of temperature, pressure and chemical potential:

1 ϑ= ^∂_ηˆ

∂e

¯

¯1 ρ,c1,...cN

, p

ϑ = ^∂_ηˆ

∂¹_ρ

¯

¯e,c1,...,cN

, −^µ_ϑ^α= ^∂_ηˆ

∂c_α

¯

¯_e,1 ρ,c_β6=α

(3.44)

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Remark1. In the definition of ˆ_η(3.39) we did not employ the constraintPN

α=1c_α=1, which follows from the definition of mass fractionsc_αin (3.38). Sometimes it is more convenient to employ this constraint when defining the specific entropy ˆ_ηand eliminate one of the concentrations (withou loss of generality c_N) and define reduced specific entropy ¯_η

η¯(e,1

ρ,c₁, . . . ,c_N₋₁)^def= _ηˆ Ã

e,1

ρ,c₁, . . . ,c_N₋₁, 1−

N−1

X

β=1

c_β

!

. (3.45)

Immediatelly, we then obtain the following relations 1

ϑ= ^∂_η¯

∂e

¯

¯1

ρ,c1,...c_N−1

, p

ϑ = ^∂_η¯

∂¹_ρ

¯

¯e,c₁,...,c_N−1

, −^µ^α−µN

ϑ = ^∂_η¯

∂c_α

¯

¯_e,1

ρ,cβ β=1,...,N−1&β6=α

. (3.46)

Note that the only difference when employing this reduction arises in the last set of relations, i.e. defining the partial derivatives of ¯_ηwith respect to reduced (independent) set of concentrationsc₁, . . . ,c_N−1, which define thedifferenceof chemical potentials with respect to the eliminated component.

Clearly one can also employ the constraint in the Gibbs relation (3.43) and get ϑd_η=d e+pd

µ1 ρ

¶

−

N−1

X

α=1

(_µ_α−µN)d c_α. (3.47)

If one defines the relative chemical potentials µ¯αdef

= µα−µN , (3.48)

one can simply use the reduced description (i.e. consider_η=_η¯³

e,¹_ρ,c₁, . . . ,c_N−1´

the Gibbs relation as

ϑd_η=d e+pd µ1

ρ

¶

−

N−1

X

α=1

µ¯_αd c_α . (3.49)

• Now, let’s consider _λ:= 1

V . Then 1-homogeneity of ˆSyields Sˆ

µE V, 1,Mα

V

¶

= 1 V

S(E,Vˆ ,Mα) . (3.50)

Recognizing in the arguments of the function on the l.h.s., the volume density of energy_ρe, and partial density_ρ_α, and observing that the function on the l.h.s. is a volume density of entropy_ρη, we can write this relation as

1 V

S(E,ˆ V,Mα)=ρη=ρηc(_ρe,_ρ₁, . . . ,_ρ_N) , (3.51) differentiating the last equality yields

d(_ρη)= ^∂ρηc

∂(_ρe)d(_ρe)+

N

X

α=1

∂ρηc

∂ρ_αd_ρ_α. (3.52)

Dividing the Gibbs-Duhem relation (3.36) byV, we get ρηd_ϑ=d p−

N

X

α=1ρ_αd_µ_α . (3.53)

Dividing the Euler relation (3.35) byV, we get ϑρη=ρe+p−

N

X

α=1

µ_αρ_α . (3.54)

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Finally, differentiating (3.54) and subtracting (3.53), we recover the local version of (3.33):

ϑd(_ρη)=d(_ρe)−

N

X

α=1

µ_αd_ρ_α . (3.55)

Note that (3.55) implies the following alternative local definitions of temperature, and chemical potential:

1

ϑ= ^∂ρηc

∂(_ρe)

¯

¯_ρ

α

, −^µ^α

ϑ = ^∂ρηc

∂ρ_α

¯

¯_ρ_e,_ρ

β6=α

(3.56) Note: One might be puzzled now, where has the pressure vanished from our description in (3.55) and (3.56). In this setting we can introduce the pressure through the Euler relation (3.54). A natural question is whether thermodynamic pressure defined in such a manner coincides with (3.44).

Exercise5. Confirm by calculation that two discussed definitions of thermodynamic pressure are com- patible. HINT: Assume that you start from fundamental thermodynamic relation_ρη=ρηc(_ρe,_ρ₁, . . . ,_ρ_N) and realize that the introduced definitions imply

ηˆ µ

e,1

ρ,c₁, . . . ,c_N

¶

= 1

MS(E,ˆ V,M1, . . . ,MN)= V M

1 V

S(E,ˆ V,M1, . . . ,MN)= V MSˆ

µE V, 1,M1

V , . . . ,MN

V

¶

=1

ρ^ρηc(_ρe,_ρ₁, . . . ,_ρ_N) .

Use this relation, define pressure through the Euler relation (3.54) and try to compute what is ϑ _∂^∂^η^ˆ1

ρ

¯

¯_e,c

1,...,cN

=ϑ_∂^∂1 ρ

¯

¯_e,c

1,...,cN

³

ρηc ρ

´

(_ρe,_ρ₁, . . . ,_ρ_N) .

3.2.2 Energetic representation

Alternatively, using the fact, that ^∂_∂^S_E^ˆ = ¹_ϑ >0, assuming sufficient smoothness of ˆS we could invert (3.27) and write instead

E=E(S,ˆ V,Mα) . (3.57)

This function was proved in 2.1 to be also 1-homogeneous, i.e.

Eˆ µ S

M, V M,Mα

M

¶

= 1

ME(S,ˆ V,Mα) . (3.58)

Differentiating this relation and comparing with (3.33), we obtained alternative definitions of absolute temperature, pressure and chemical potential:

ϑ=^∂ Eˆ

∂S , p= −^∂ Eˆ

∂V , _µ_α= ^∂ Eˆ

∂Mα . (3.59)

Both the Euler relation (3.35) and the Gibbs-Duhem relation (3.36) are recovered in the same form, and one can proceed to the local forms as before, by applying the 1-homogeneity w.r.t total mass and total volume. The same form of local Euler and Gibbs-Duhem relations are recovered, plus we obtain the following two sets of alternative definitions:

ϑ= ^∂eˆ

∂η

¯

¯₁

ρ,c1,...cN

, −p= ^∂eˆ

∂¹_ρ

¯

¯η,c1,...,cN

, _µ_α= ^∂eˆ

∂c_α

¯

¯_η_,1 ρ,c_β6=α

(3.60)

for

e=e(ˆ_η,1

ρ,c₁, . . . ,c_N) , (3.61)

and

ϑ= ^∂^ρce

∂(_ρη)

¯

¯ρα

, _µ_α= ^∂^ρce

∂ρ_α

¯

¯ρe,ρβ6=α

(3.62) for

ρe=ρce(_ρη,_ρ₁, . . . ,_ρ_N) . (3.63)

T HEORY OF M IXTURES