A nonlinear coupling of chemical kinetics with mechanics V. Klika 1 , - PowerPoint PPT Presentation

A nonlinear coupling of chemical kinetics with mechanics V. Klika 1 , 2 , M. Grmela 3 1 Institute of Thermomechanics of the AS CR, CR klika@it.cas.cz 2 Dept. of Mathematics, FNSPE, Czech Technical University in Prague, CR 3 Chemical Engineering, Ecole Polytechnique, Montreal, Canada 23.8.12 Roros, IWNET 12 Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 1 / 25

Outline Motivation 1 A first approximation of coupling - CIT 2 A non-linear mechano-chemical coupling - GENERIC 3 A feedback to motivation? 4 Conclusions 5 Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 2 / 25

Motivation Bone and BR Why are we interested? the importance of mechanics (coupling) two groups of models (lack of communication) bridging them with NET Functions of BR in bone:to keep bone alive, to alter the shape of bone, repair damages in bone tissue, part of metabolism Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 3 / 25

A first approximation of coupling - CIT Finding model formulation I For interaction schemes � k ± α � ′ ν α j N j α j N j , α = 1 , .., s ⇄ ν j j the Law of Mass Action is used (with some limitations) n n � � ′ [ N j ] ν α j − k − α [ N j ] ν r α = k + α α j j =1 j =1 Consequently, the change in concetration of subst. in time � ˙ ′ [ N j ] = ( ν α j − ν α j ) r α α where j refers to a substance N j , r α is the rate of α − th interaction, ν α j is stoichiometric coefficient for entering, outcomming substances N j of interaction α , respectivelly Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 4 / 25

A first approximation of coupling - CIT Finding model formulation of II Phenomenological relations Entropy production � � n s � � σ ( S ) = j q ∇ 1 + 1 T t dis : ∇ v + 1 ∇ µ i T − F i T − j D i r α A α ≥ 0 . T T i =1 α =1 and its general form: σ ( S ) = J s X s + J v · X v + J a a · X a a + J t : X t . CIT for a scalar (rate of chemical reaction): ◦ ◦ J s = L ss X s + L sv · X v + L a sa · X a a + L st ( s ) : X t ( s ) . WLOG L st is of the same kind as the thermodynamic force X t (thus, ◦ L st = L st ( s ) is a symmetric tensor with zero trace) Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 5 / 25

A first approximation of coupling - CIT Rate of deformation tensor and entropy production � ∂ ˙ � � ∂ v i � u i u j ∂ x j + ∂ v j � ( ∇ v ) ij + ( ∇ v ) ji � E ij = 1 ∂ x j + ∂ ˙ = 1 = 1 D ij = ˙ , ∂ x i ∂ x i 2 2 2 and for the rate of volume variation D (1) it holds: E (1) = div v = − 1 d ρ D (1) = ˙ ρ d t note: tensor D is the symmetric part of tensor ∇ v and thus s � T σ ( S ) = t dis : ∇ v + r α A α = α =1 � �� s � � � � � � � � � ◦ � r α A α +1 ◦ = 3 tr ∇ v tr t dis + ∇ v ( a ) · t dis ( a ) + D : t dis ( s ) ≥ 0 , α =1 where ∇ v was decomposed into scaled unit tensor and a symmetric and an antisymmetric parts with zero traces. Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 6 / 25

A first approximation of coupling - CIT The isotropic system To recall: ◦ ◦ J s = L ss X s + L sv · X v + L a sa · X a a + L st ( s ) : X t ( s ) . Constraints on phenomenological coefficients follow (from invariance of isotropic system under orthogonal transformations - does not modify the phenomenological tensors): inversion ( L sv = 0), arbitrary rotation ( L a sa = 0) Finally, from the fact that scalar quantity is not affected by orthogonal transf, namely a T b it follows L st = L st U . Moreover, tr L st = 0 ⇒ L st = 0. In total (Curie principle): J s = L ss X s or in particular r α = L s 1 A α + L s 2 D (1) . Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 7 / 25

A first approximation of coupling - CIT Other choices of decomposition - scalar quantity D (2) Analogous procedure; difference - tensors are not traceless Still constraints on phenomenological coefficients follow (from invariance of isotropic system under orthogonal transformations - does not modify the phenomenological tensors): inversion ( L sv = 0), arbitrary rotation ( L a sa = 0) but L st � = 0, only L st = L st U , In total, CIT and Curie-Prigogine principle leads to (notice the dynamic origin; static l. when viscous effects are significant but still through strain rate) r α = L s 1 A α + L s 2 D (1) = L s 1 A α + L s 2 D (2) + L s 3 D (1) . Klika, J Phys Chem B . 2010 Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 8 / 25

A first approximation of coupling - CIT Finding model formulation III Modified Law of Mass Action The most important (least worst option) invariant is D (1) and thus ⇒ r α = l αα A α + l α v D (1) = Expressions of affinity and chemical potential in CIT lead to 1 The Law of Mass Action (when coupling neglected) n n � � ′ [ N j ] ν α i − k − α [ N j ] ν r α = l αα A α = k + α α i j =1 i =1 2 The modified Law of Mass Action (including mechano-chemical coupling) n n � � r α = l αα A α + l α v D (1) = k + α ′ [ N j ] ν α i − k − α α i + l α v D (1) [ N j ] ν j =1 i =1 ( C ijkl = C ijkl ( N j ) = C ijkl � � N j ( D (1) ) ) Klika, Marˇ s´ ık, J Phys Chem B . 2009 Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 9 / 25

A non-linear mechano-chemical coupling - GENERIC Prerequisities Isothermal - φ instead of S Mechanical scalar variables (coupling depends on parity): even-parity a , and odd-parity b Chemical variables: molar concentrations n and their fluxes z Conjugates ( a ∗ , b ∗ ) , ( n ∗ , z ∗ ), a ∗ = ∂ a φ Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 10 / 25

A non-linear mechano-chemical coupling - GENERIC Uncoupled mechanics The time evolution of simplified but complete mechanics relevant for coupling (GENERIC): � ˙ � � � � a ∗ � � � a 0 0 κ = − ˙ − κ T Θ mech 0 b ∗ b b ∗ � � � � κ b ∗ 0 = − . − κ T a ∗ Θ mech b ∗ The dissipation potential Θ mech ( a , b ∗ ) satisfies: Θ mech is a real valued and sufficiently regular function of ( a , b , b ∗ ) Θ mech ( a , b , 0) = 0 Θ mech ( a , b , b ∗ ) reaches its minimum at b ∗ = 0 Θ mech ( a , b , b ∗ ) is a convex function of b ∗ in a neighbourhood of b ∗ = 0 . Θ mech is usually considered quadratic. Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 11 / 25

A non-linear mechano-chemical coupling - GENERIC Notes on structure of the evolution eq � ˙ � � � � a ∗ � � � a 0 0 κ = − ˙ − κ T Θ mech b 0 b ∗ b ∗ � � � � κ b ∗ 0 = − , − κ T a ∗ Θ mech b ∗ The first term : time reversibility (invariance under t → − t and change of parity). The second term : in mechanics, dissipation is typically a friction ∝ lin momenta b ⇒ dissipation term has odd-parity; ⇒ dissipation term is expected to be only in the evolution of linear momentum b ; We let only the odd-parity variables dissipate Note that d Φ dt ≤ 0 (consistency with thermodynamics). Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 12 / 25

A non-linear mechano-chemical coupling - GENERIC Uncoupled chemisty, LoMA LoMA (Guldberg-Waage) in GENERIC form (yesterday): n = − Ξ chem ˙ n ∗ with dissipation potential � � s � e − 1 2 X ρ + e 1 2 X ρ − 2 Ξ chem ( n , X ) = W ρ ( n ) ρ =1 where thermodynamic forces X = ( X 1 , ..., X s ) T are K � γ k ρ Φ chem X ρ = n k k =1 together with the natural choice of entropy form S ( n ) = − � K j =1 ( n j ln n j + Q j n j ) where Q 1 , ..., Q K can be calculated. Note that n ∗ = µ . Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 13 / 25

A non-linear mechano-chemical coupling - GENERIC Uncoupled chemisty, Extended Fluxes (odd-parity momentum-like quantity) considered as independent variables (motivated by EIT; yesterday) From the analogy with mechanics � ˙ � � � � n ∗ � � � n 0 0 γ = − − γ T Θ chem ˙ 0 z z ∗ z ∗ � � � � γ z ∗ 0 = − − γ T n ∗ Θ chem , z ∗ Note that the reversible part of the right-hand-side of ˙ z is equal to affinity. The standard LoMA formulation in GENERIC is required when fluxes equilibrate (separation of timescales) → disip pot Θ chem . Now n acquire dissipation. Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 14 / 25

A non-linear mechano-chemical coupling - GENERIC Coupled chemical kinetics and mechanics         0 n ˙ 0 0 n ∗ γ µ         0 ˙ a 0 0 a ∗   ν κ       =  −   Θ (1 , n , a , z , b )      − γ T − ν T ˙ z 0 0 z ∗   z ∗ ˙ − µ T − κ T Θ (2 , n , a , z , b ) 0 0 b ∗ b b ∗   γ z ∗ + µ b ∗ κ b ∗ + ν z ∗     =   − γ T n ∗ − ν T a ∗ − Θ (1 , n , a , z , b )   z ∗ − κ T a ∗ − µ T n ∗ − Θ (2 , n , a , z , b ) b ∗ Again dissipation is assumed only in evolution of odd-parity variables and reversible evolution of a state variable is caused by conjugate state variables with different parity Klika,Grmela () Chemo-Mechanical Coupling IWNET 12 15 / 25