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A New Formulation and Computation of the Triphasic Model for Mechano-electrochemical Mixtures Y.C. Hon1 M.W. Lu 2 W.M. Xue3 X. Zhou 4 1Department of Mathematics, City University of Hong Kong, Hong Kong

2Department of Engineering Mechanics, Tsinghua University, Beijing 100084, PRC. 3Department of Mathematics, Hong Kong Baptist University, Hong Kong 4 Beijing Special Engineering Design and Research Institute, Beijing 100028, PRC.

ABSTRACT From the generalized rst law of thermodynamics for an irreversible thermodynamical system, a new set of governing equations for the mixture theory is derived based on the triphasic model for mechano-electrochemical mixtures. It is shown that, in the case of electroneutral solution, a new biphasic mixture theory including the electrochemical eects can be derived from the new governing equations. The chemical-expansion stress representing both the in uences of deformation on the xed charge density and the electric potential of xed charge eld is given. For comparison and veri cation purposes, the numerical solution for a con ned compression problem of a charged hydrated soft tissue is computed using the multiquadric method.

INTRODUCTION Biological tissues are multiphasic materials consisting of various amounts of living cells, extracellular matrix, and interstitial uid. To a large extent, the biorheological properties of connective tissue are dependent upon the compositional and ultrastructural properties of the extracellular matrix. For soft connective tissues, the extracellular matrix is 1

composed of a collagenous brous network which is swollen by a highly hydrophilic glycosaminoglycan gel. All connective tissues exhibit some viscoelastic behaviors to various degrees. Some of these apparent viscoelasticity is ascribable to the intrinsic viscoelasticity of the matrix macromolecules, namely, collagen and glycosaminoglycan. Biological tissues consist of a large proportion of water. Much of this water is movable and could move as a result of deformation of the tissue matrix. The frictional drag as the interstitial uid moves relates to the tissue matrix has been regarded as one of the most important mechanisms of energy dissipation during tissue deformation. The physics of interstitial ows is important not only in tissue rheology, but also in other electromechanical phenomena as the electrolytic interstitial uid carrying mobile ions/charges moves relative to the tissue matrix with xed charges, as well as in other chemomechanical phenomena during the concomitant movement of the interstitial water as small mobile ions move down a concentration gradient. This understanding is important to our appreciation of the transport mechanisms in biological tissues as well as the transduction mechanisms of physical stimuli in the micro-environment within the tissue. In this paper, a new set of governing equations for the triphasic mixture theory based on the triphasic model proposed by Lai et al. [1] is derived from the generalized rst law of thermodynamics for an irreversible thermodynamical system. In the newly derived model, the ionic concentration is expressed by the chemical energy and the chemical expansion stress is represented by the electrostatic energy. The eect of the diusive resistance is then included in the energy dissipation. This greatly simpli es the governing equations for the triphasic model given by Lai et al. [1]. For an electroneutral solution, i.e. in absence of electric current, the new governing equations can be simpli ed to a new biphasic mixture model which includes the electrochemical eects. For comparison and veri cation purposes, the numerical solution for a one-dimensional con ned compression problem of a charged hydrated soft tissue is computed using the multiquadric (MQ) method which was developed by Hon et al. [12] for solving the biphasic mixture model. PRELIMINARY ASSUMPTIONS AND NOTATIONS IN TRIPHASIC MODEL Hydrated biological soft tissues such as articular cartilage have been idealized by a triphasic model [1, 2], which consists of three phases: (1) a solid phase including the collagen bres, proteoglycan aggregates (PGA) with its xed negative charges, superscribed by s, (2) an interstitial water phase, superscribed by w, and (3) an ion phase of dissolved salt 2

consisting of cations (Na+ ) and anions (Cl? ), superscribed by i. Each phase is assumed to be intrinsically incompressible. The solution, which consists of solvent water and solute ions, is called uid phase, superscribed by f (f = w [ i). It is assumed that the solution is electroneutral and no electric current occurs. The electrostatic eect of xed charge groups along the glycosaminoglycan (GAG) chains is considered in this paper. Density and concentration.

The volume fraction of each phase  ( = s; w; i) is given by  V = V ( = s; w; i) (1) in which V  is the true volume of phase , V is the mixture volume, which is equal to the apparent volume of each phase , ( = s; w; i) satisfy the continuity condition of mixture: X  = 1: (2)



=s;w;i

The apparent mass density of phase  ( = s; w; i) is

 = T ( = s; w; i);

(3)

where T is the true mass density of phase . The total density of the mixture, i.e. the tissue, is X  = X  : (4) = T =s;w;i

=s;w;i

In the study of the chemical eects, the molar concentration per unit solution ( uid phase) volume,  c = fM  ( = s; w; i); (5)

is used. Here, M  is the molecular weight of species  and i is assumed to be comparatively very small so that f = w + i  w . Basically there are two mechanisms for tissue swelling eect resulted from the ionic phase. The rst is the well known Donnan osmotic pressure caused by the concentration dierence between the interstitial solution and the external bathing solution. The other is the so{called chemical{expansion stress eected by the electrostatic repulsive forces of the xed charges. It is assumed that the number of xed charges per unit mass of solid 3

phase is kept constant at any time [1]. The molar concentration of the negative xed charges based on the apparent volume V of the solid phase is then de ned by: F F cF = dn = c (6) 0 J; dV where nF is the total molar number of xed charges, cF0 = dnF =dV0 is the value of cF at the reference con guration. The volume ratio of apparent solid phase, J = dV0=dV , is related to the Green strain tensor E of the apparent solid phase as follows: 1 = q1 + 2J (E) + 4J (E) + 8J (E)  1 + J (E); (7) 1 2 3 1 J where J1(E) = tr(E), J2(E), and J3(E) are the rst, second, and third invariant of Green strain tensor E. We note here that the element of the triphasic mixture is referred to the porous solid matrix, i.e., the element of the apparent solid phase, so that J s = J , Es = E. For in nitesimal deformation, we obtain from (7) that

J = 1 ? tr(E):

(8)

Substituting equation (8) into (6) we have

cF = cF0 (1 ? tr(E)):

(9)

The xed negative charges on the GAG attract mobile cations Na+ to maintain the macroscopic electroneutral state. By assuming that each negative xed charge attracts a cation Na+, the molar concentration of the attracted cations Na+ per unit uid volume can be de ned as dnF = cF ; ca = dV (10) f f

and for in nitesimal deformation,

f = 1 ? s0J = f0 (1 + f0 tr(E)): 0 s

(11)

Electroneutrality.

Denote the concentration of free cations (Na+)f and anions Cl? in a solution of electroneutral salt per unit uid volume by c+f = c? = ci. Due to the presence of additional attracted cations, the concentration of Na+ is

c+ = c+f + ca = ci + ca: 4

(12)

The electric current density, denoted by I , is calculated by

I = FC w [c+f (v+ ? vs) + ca(va ? vs) ? c?(v? ? vs)];

(13)

where FC = 96485C/mol is the Faraday constant and v+; va; v?, and vs are the velocities of the free cation, attracted cation, anion, and solid matrix respectively. Due to the absence of electric current and the fact that the velocity of attracted cations is macroscopically equal to vs, equation (13) becomes 0 = FC w [ci(v+ ? vs) + ca(vs ? vs) ? ci(v? ? vs)]: Hence,

v+ = v?  vi:

(14) In other words, if the electric current density in an electroneutral solution is zero, the ion phase can be treated as an uniform phase for the consideration of mechanical eects such as movement and deformation.

Continuity of mixture consisting of incompressible components. The velocities vs; vw and vi of phases s; w; and i respectively must satisfy the continuity condition. The continuity equation for each phase  is

i.e.

D = ? 5
v Dt

( = s; w; i);

(15)

@ + 5
(v) = 0 ( = s; w; i): (16) @t Substituting (3) into (16) and using the assumption for intrinsic incompressible condition, T = constant, we obtain @ + 5
(v) = 0 ( = s; w; i): (17) @t The three equations in (17) and the continuity condition of mixture (Eq. 2) can be combined to X v) = 0: (18) 5
( Using the relations of tensor analysis:

=s;w;i

5
(v) =  5
v + v
5 ( = s; w; i); 5

(19)

5
v = I : 5v = I : d ( = s; w; i);

(20)

we obtain from equation (18) the continuity equation of the mixture consisting of incompressible components: X (I : d + v
5) = 0; (21) =s;w;i

where I is the identity tensor and d is the deformation rate tensor which is a symmetric component of the velocity gradient tensor 5v. ENERGY, WORK, HEAT AND DISSIPATION IN TRIPHASIC MIXTURES Consider a triphasic mechano-electrochemical mixture system in which the total energy, work, heat and dissipation of the mixture is expressed in a summation form of its value in each phase consisting of the mixture. The rate of kinetic energy K_ of the triphasic mixture is given by

K_ =

X K_  = Z ( X v
v_ ) dV; V

=s;w;i

=s;w;i

(22)

where v and v_  are the velocity and acceleration of the  phase respectively. The notation (_) represents a material derivative. The rate of internal energy U_ of the triphasic mechano-electrochemical mixture consists of three portions:

U_ =

Z

X (U_  + C_ + E_ ) dV;

V =s;w;i

(23)

where U_ , C_ and E_ are the rates of strain energy density, chemical energy density and electrochemical energy density of  phase per unit mass respectively. We introduce a Helmholtz energy function F :

F = U ? TS;

(24)

where T and S are the absolute temperature and entropy of the considering system. The density of Helmholtz energy and entropy per unit mass of phase  is denoted by F  and  respectively. By taking derivative of equation (24), we obtain

F_ =

Z

V

F_  dV =

Z

X F_  dV

V =s;w;i

6

=

Z

X (U_  + C_ + E_ ? T_  ? T _ ) dV;

V =s;w;i (F_ )= represents

(25)

where F_ = P=s;w;i the Helmholtz energy density of the mixture. The same terminology without superscript  is also applied to U_ , C_, and E_ . Both internal energy U and Helmholtz energy F are state functions depending on their state variables. For the triphasic mixture system, the following parameters are selected as the independent variables. These are the absolute temperature T as a thermal parameter, the Green strain tensor E as a mechanical parameter which is reduced to the Euler strain tensor in the case of in nitesimal deformation, the apparent densities  ( = s; w; i), which are related to the concentrations, as the chemical parameters, and the molar concentration of xed charges cF as a electrochemical parameter. The Helmholtz energy density can then be expressed as F  = F (T; E; s; w ; i; cF ) ( = s; w; i): (26)

Here, the Green strain tensor E of the apparent solid phase is selected as one of the state variables. The deformation eect of other phases will be considered by introducing the continuity equation (21) consisting of incompressible components. For the chemical energy density C in (25) we consider only the chemical eect of a mass transfer process, which is related to the concentration of species, but not the chemical reactions in which the mass of species are generated or exhausted. The chemical reaction is important for biological phenomena of the respiratory system and the digestive system, etc. However, it may be neglected for the deformation of passive biological tissues like cartilage [5]. Hence, the molar chemical potential ^ is given by

@F = @ F~ = @ C~ ( = s; w; i); ^ = @n  @ c~ @ c~

(27)

where c~, F~ = F and C~ = C are the molar concentration, Helmholtz energy and chemical energy per unit mixture volume respectively. n is the total molar number of species . The last equality of (27) due to the U ; E and T in (25) are irrelevant to the chemical eect caused by the concentration dierence. The chemical potential density  is de ned to be: @ F~ = @ F~ @ c~ = ^ ( = s; w; i);  = @ (28)  @ c~ @ M  where M  is the molecular weight of species . For a real solution at constant temperature 7

and pressure, the chemical potential density of species  is equal to RT ln ( c) ( = s; w; i);  = 0 + M (29)  where the constant 0 represents the chemical potential density of the reference state, R is the universal gas constant,  is the activity coecient, which is equal to 1.0 for an ideal solution, and c is the molar concentration per uid volume. Chemical potential of the solid phase of articular cartilage comes mainly from the macromolecules of the proteoglycan aggregates (PGA), whose quantity is approximately 10% W/W or 7% V/V of the mixture. The in uence of collagen to chemical potential and osmotic pressure can be negligible [6]. A theoretical treatment on chemical potential and osmotic pressure of biological macromolecules is given by Tombs and Peacocke [7]. For the electro-chemical energy density E in (25) we consider only the electrostatic eect of the xed charges. In the absence of electric current, the electro{chemical eect related to the ow of mobile ions in an electroneutral solution may be neglected. The negative xed charge groups (SO3? and COO? ) along the GAG chains of proteoglycan aggregates (PGA) in the biological tissues are distributed very close to each other, spaced at 10 to 15 Angstroms apart [8] [9]. When the mixture deforms, the distance between the xed charges changes. This results in a variation of the eletrostatic potential, which is related to the chemical-expansion stress TC . The electrostatic potential at point P caused by a xed charge distribution qV , which is considered to be composed of point charges, is ~ Z = @ E = qV dV; (30) @qV V 4"r where E~ = E is the electro-chemical energy per unit mixture volume, " is the permittivity and r is the distance from point P to the xed charge distribution qV . Although the exact distribution of xed charges is unknown, it is known that the electrostatic potential at P is mainly determined by its surrounding charges. We can assume that the distribution of the xed charges in the neighborhood of P keeps its pattern during the deformation. The distance r is then given by (31) r = r0(1 + 13 tr(E)): Here tr(E) takes its value at P . Using qV = FC cF0 and substituting equation (31) into (30) and integrating over a neighborhood region V of the point P , we obtain Z qV 1 (32) = 1 + 1 tr(E) 4"r dV  0(1 ? 13 tr(E)); V 0 3 8

where

Z FC cF0 dV 0= V 4"r0

(33) is the electrostatic potential of xed charge eld at the reference state. When the ion concentration of solution c increases, the permittivity " also increases due to the charge shielding eect caused by the ion cloud moving into the space between two repulsing xed charges. This results in an exponential decrease of electrostatic potential 0: a0 exp(?k c); (34) 0= FC where FC is the Faraday constant for the charge value of 1 mol charges, a0 and k are two experimental material constants. From equations (30), (32) and (34) we obtain the molar eletrostatic potential of the xed charge eld F as follows: @ E~ = F = a exp(?k c)(1 ? 1 tr(E)): (35) F  @c C 0 F 3 The rate of works W_ e done by the external forces is given by Z Z _We = X f f 
v dV + t
v dS g; (36) =s;w;i

V

S

where f  is the body force per unit mass, t = 
 is the drag force applied on the surface element,  is the Cauchy stress tensor, and  is the external normal on the surface element. From the Gaussian gradient formula and the symmetry of stress tensor, the surface integration in (36) can be transformed into a volume integral as follows: For  = s; w; i,

Z


v dS = t S

Z

S

(
v)
dS =

Z

Z

5
(
v)dV = [(5
)
v +  : 5v]dV: V V

Substituting into (36) we obtain Z X [(5
 + f )
v +  : 5v]dV: W_ e = V =s;w;i

(37)

The rate of work done by the pressure p is ?pV_ . However, in the incompressible case V_ = 0, it may be computed by the right hand side of the continuity equation (21) consisting of incompressible components: Z X (I : d + v
5) dV: (38) W_ p = ? p V

=s;w;i

9

The total rate of work is then equal to

W_ = W_ e + W_ p:

(39)

The rate of heat transfer Q_ into the system is de ned as

Q_ =

X fZ   dV ? Z q
 dS g; S V

=s;w;i

(40)

where  is the rate of heat generation per unit mass of phase , which may include the heat generation of chemical reaction, and q is the heat ux vector. Again, by using the Gaussian gradient formula we obtain

Q_ = ?

Z X (5
q ?  ) dV: V =s;w;i

(41)

The energy dissipation can also be considered in the energy formulation of irreversible thermodynamics [10]. In the triphasic mixture theory for articular cartilage and biological tissues, the dissipation is mainly caused by the diusive resistance due to the relative

ow between two dierent phases  and . We assume that the diusive drag  is proportional to their relative velocity:

 = X K (v ? v); =s;w;i

(42)

where K = K  is called diusive drag coecient and represents other phases which is dierent from  in the mixture. It follows from (42) that

X   = 0:

=s;w;i

(43)

The rate of energy dissipation D_ is de ned as

D_ =

X Z 
v dV: V

=s;w;i

(44)

For a biphasic mixture consisting of solid phase and uid phase, equation (44) leads to

D_ = D_ s + D_ f = ?

Z

V

Ksf (vf ? vs)
(vf ? vs)dV:

It follows that the rate of energy dissipation D_ is always a negative function. GOVERNING EQUATIONS OF TRIPHASIC MODEL 10

(45)

The governing equations of triphasic model can now be derived from a generalization of the rst law of thermodynamics. The balance relation on the rate of energy including the dissipation for an irreversible thermodynamical system is given by _ K_ + U_ ? D_ = W_ + Q:

(46)

Substituting equations (22), (23), (25), (39), (41) and (44) into (46), we derive that

Z

X [v_ 
v + (F_  + T _  + T _ ) ? 
v ? p 5 
v ]dV

V =s;w;i

=

Z X f [(5
 + f )
v +  : 5v] ? pI : d ? (5
q ?  )gdV: V =s;w;i

The material derivative of the Helmholtz function (26) is: For  = s; w; i, F  T_ + @ F  E_ + X @ F  _ + @ F  c_F : F_ (T; E; s; w ; i; cF ) = @@T @E @cF =s;w;i @ From the continuity equations (15) and (20) we have _  D Dt = ? I : d ( = s; w; i): From equation (6) and the fact that J_ = ?J 5
vs we obtain

c_F = ?cF0 J I : ds: Notice that

(47) (48)

(49) (50)

E_ = (Fs )T
ds
Fs ;

(51) where Fs and (Fs )T are the deformation gradient tensor and its transpose respectively. Using equation (51) we have @ F  : E_ = @ F  : ( (Fs )T
ds
Fs) = (Fs
@ F 
(Fs )T ) : ds: (52) @E @E @E Substituting equations (49), (50) and (52) into (48) we obtain the formula for the material derivatives of the Helmholtz function: For  = s; w; i, F  T_ + (Fs
@ F 
(Fs )T ? cF J @ F  I) : ds ? X  @ F  I : d : (53) F_  = @@T 0 @cF @E @ =s;w;i Using

 : 5v =  : d;

11

(54)

and substituting equation (53) into (47) we have:

Z X [5
 + f  ? v_  +  + p 5 ]
v dV V =s;w;i Z X @ F~  )
(Fs )T ? cF J ( X @ F~  )I ] : ds + f[Fs
( 0 F V =s;w;i @ E =s;w;i @c ~ X X @ F    + [? ? p I ? (  )I ] : dgdV

?

=s;w;i

@ =s;w;i  + T_  ?   ]dV

Z X + [5
q V =s;w;i Z X @ F~    +   ]T_ dV = 0; [ + V =s;w;i @T  when  = s.

(55)

where ds is a component of d Since the rates v; d; and T_ change independently, all of the terms contained in each square bracket of equation (55) should be equal to zero. A set of governing equations for the triphasic mixture model can now be derived. Before listing down all the equations, the physical meaning of some terms in equation (55) is discussed in the following. Using equation (28) the chemical term becomes X  @ F~ =  @ (P F~ ) =  @ F~ =  ( = s; w; i): (56) @ @ @ =s;w;i The electrostatic term in equation (55) is related to the chemical-expansion stress TC which appeared in the triphasic model developed by Lai et al. [1] but was not fully understood. Since the other components of F are irrelevant to the electrostatic eect, we have @ F~ = X @ F~  = @ E~ : (57) @cF @cF @cF =s;w;i

From equations (8),(57), and (35) we derive that X @ F~  = T (1 ? tr(E))(1 ? 1 tr(E)); TC (cF0 ; c; E)  cF0 J C0 F 3 =s;w;i @c = TC0 (1 ? 43 tr(E)); and TC0 (cF0 ; c) = a0cF0 exp(?kc): 12

(58)

(59)

It is now clear that the chemical-expansion stress TC is related to the concentration of xed charges, the ion concentration of solution and also the in uences of deformation. The factors (1 ? tr(E)) and (1 ? 13 tr(E)) represent the in uences of deformation on the xed charge density and the electric potential of xed charge eld respectively. The mechanical term in equation (55) is corresponding to the stress tensor of solid phase, in which X @ F~  = @ F~ =  s @E @E E =s;w;i

is the second Piola-Kirchho stress tensor, and

Fs
 sE
(Fs)T = sE

(60)

is the Cauchy stress tensor. Finally, by substituting equations (54), (56), (58), and (60) into (55), and equating the terms contained in each square bracket in equation (55) to zero, we obtain the following governing equations for triphasic mixtures: Momentum equations

5
 + f  ? v_  +  + p 5  = 0

( = s; w; i);

(61)

Constitutive equations  s = ?s pI +  sE ? s s I ? TC I   = ?pI ?   I

( for solid phase ); ( for  = w; i );

(62) (63)

For isotropic elastic material, sE is given by  sE = s tr(E)I + 2s E;

(64)

where s and s are the Lame elasticity constants. Heat transfer equation

5
q + T_  ?   = 0 ( = s; w; i): A de nition for the entropy density  can also be obtained from (55) : F~  ( = s; w; i):  = ? 1 @@T 13

(65)

(66)

From equations (2) and (43), the three equations for the three phases s; w; i in (61) can be summarized as a total momentum equation for the triphasic mixture model:

5
 + f ? v_ = 0;

(67)

where  = P=s;w;i , f = P=s;w;i f =, and the density average velocity v is de ned by (68) v = ( X v)=: =s;w;i

Equations (62){(64) governs the mechano-electrochemical coupled constitutive relations. From the derivation of equation (55), it can be observed that the coupling eect, including the chemical term  and electrostatic term TC , comes from the volume, and therefore the density and concentration variation. For the case of electroneutral solution, the following derives a new biphasic formulation of the triphasic mixtures by treating the uid phase as the water phase and the ion phase as an uniform electrolytic solution phase. A NEW BIPHASIC FORMULATION OF THE TRIPHASIC MIXTURES Based on the conditions of electroneutrality and an absence of electric current, the ion phase including the cations and the anions can be considered as an uniform electroneutral salt phase, the velocity of which is denoted by vi (see equation (14) ). Furthermore, due to the absence of external electric eld, there exists no driving force pushing the ions to move macroscopically dierent from the water. It may then be assumed that

vi = vw  vf :

(69)

This means that the electroneutral salt and the water can be considered as an uniform electrolytic solution phase from the view point of the motion and deformation. The triphasic model derived in the previous section can then be simpli ed to a mechanoelectrochemical biphasic mixture model, which consists of a porous solid phase and an uniform electrolytic uid phase. The basic equations of this new biphasic mixture model can be derived directly from the above governing equations of the triphasic mixture model. By composing the equations for the water and ion phases together, equations (61){(63) and (16) are reformulated into the following governing equations for mechano-electrochemical biphasic mixture model: 14

Momentum equations

r
s + s f s ? s v_ s + s + pr's = 0 r
f + f f f ? f v_ f + f + pr'f = 0

( = s); ( = f = w [ i);

(70) (71)

From equation (42) and the assumption vw = vi  vf we have

f = Ksf (vf ? vs) = ?s; where

(72)

Ksf = Ksw + Ksi:

Constitutive equations  s = ?'s pI +  sE ? (s s + Tc )I f

where

( = s); ( = f );

= f I = ?'f pI ? f f I  sE

= s tr(E)I + 2s E:

(73) (74) (75)

Continuity equations

where

@'s + r
('s vs) = 0 @t @'f + r
('f vf ) = 0 @t

( = s);

(76)

( = f );

(77)

'f = 'w + 'i  'w ; f = w + i  w ; f = w + i; f f = (w f w + i f i)=f ; vf = (w vw + i vi)=f = vw = vi; f = (w w + i i)=f ;

f = w + i;

(78) (79) (80) (81)

with the approximation sign \" denoted for dilute solution. For an osmotic process at constant temperature, the chemical potential and pressure change with respect to the concentration variation due to osmosis. Hence, they satisfy the Gibbs-Duhem equation: sds + f df = dp: (82) 15

For dilute solution, integrating equation (82) we have

s(s ? s0) + f (f ? f0 ) = posm :

(83)

where s0 and f0 are the chemical potentials at the reference state, posm is the osmotic pressure. The gradient of (83) is

r(ss ) + r(f f ) = rposm :

(84)

By summarizing the equations of the solid phase and the uid phase, and using equations (2), (72), (83) and (84), we obtain the following basic equations for the electrochemical biphasic mixtures: Momentum equation r
 + f ? v_ = 0; (85) in which rposm is included in the gradient of the stress tensor. Constitutive equation  =  sE ? (p + Tc )I; (86) in which posm is included in the pressure p. Continuity equation r
('svs + 'f vf ) = 0: (87) Substituting the constitutive equations (73) and (74) into (70) and (71) we obtain:

?'srp + r
sE + s ? r(ss + Tc) + f s ? s v_ s = 0 ?'f rp + f ? r(f f ) + f f ? f v_ f = 0 for uid:

for solid;

(88) (89)

Using equation (84) and the relation of partial pressure and substituting (86) into (85), we obtain: Mechano-electrochemical coupled momentum equations:

?'srp + r
sE + s ? rTc + f s ? sv_ s = 0 for solid; ?'f rp + f + f f ? f v_ f = 0 for uid; r
sE ? r(p + Tc) + f ? v_ = 0 for mixture:

(90) (91) (92)

in which posm is included in the pressure p. To further understand the eect of the ion phase in the triphasic model, the above mechano-electrochemical biphasic model is compared with the classical one developed by Mow and Lai [3]{[5]. The result of comparison is listed as follows: 16

(I) the momentum equations (70), (71), and (85) and the continuity equations (76), (77), and (87) are the same as the classical biphasic model. Two dierences in [5] are: (i) the terms pr' ( = s; f ) in the continuity equations are merged into the diusive drag ; and (ii) the body forces f  and the inertial forces ?v_  are neglected. (II) The electrochemical eects are embodied only in the constitutive equations. If the chemical term  and the electrostatic term Tc are neglected, equations (73), (74), and (86) can be reduced to the classical biphasic constitutive equations. (III) Equations (73) and (74) indicate that from the view point of mechanics, the chemical potential multiplied by the mass density and the electrostatic potential multiplied by the charge concentration can be considered as a generalized stress, which is an isotropic stress tensor similar to the pressure. (IV) The true stress tensors are obtained from equations (73) and (74) by dividing a volume fraction ': s T  sT = Es ? (p + sT s + cs )I; (93) ' ' (94)  fT = Tf I = ?(p + fT f )I: Dividing the scalar coecient of equation (94) by ?fT , we obtain f  T (95)  ? f = pf + f ; T T where f is the net chemical potential caused by the concentration variation of species and f is a generalized chemical potential including the eect of pressure. From the viewpoint of chemists, the true stress tensor Es ='s in (93) divided by the solid density sT can be considered as a non-isotropic chemical potential tensor of deformed solid phase. (V) Neglecting the body force and the inertial force, we reformulate equation (89) into

f

f rf + f rf + 'f rp = Ksf (vf ? vs):

(96)

It can be seen that the gradients of chemical potential, pressure and concentration are the driving forces to push the uid to ow and to overcome the diusion resistance. NUMERICAL COMPUTATION To verify and compare the newly developed triphasic mixture model with the biphasic model, the numerical solution for a one-dimensional con ned compression problem of 17

charged hydrated soft tissue with similar settings given by Hon et al. [12] is given. The proposed multiquadric (MQ) method in [12] is used so that the original computer program for solving the biphasic model can easily be modi ed to obtain the numerical solutions. Consider a specimen of charged hydrated soft tissue placed snugly inside a circular con ning chamber (Fig. 1). The sample is equilibrated in a NaCl solution. Before the applying of the ramp displacement, the tissue is in a swollen state, which is taken as the reference con guration. The history of the ramp displacement load is illustrated in Fig. 2. Governing Equations.

In one-dimensional case, the continuity equation (87) can be written as @ (svs + f vf ) = 0: (97) @z Substituting s = 1 ? f into equation (97), we obtain @vs = @ [f (vs ? vf )]: (98) @z @z Since the velocities vs and vf are zeroes at z = 0 for all time, the integration of equation (98) with respect to z gives @us = f (vs ? vf ); (99) @t where us is the displacement of the solid phase. If the body force and the inertial force are neglected, the momentum equations (90) and (91) can be written as @p + K (vs ? vf ) = 0; ?f @z (100) sf 2us @Tc @p @ s ? @z + (s + 2s ) @z2 ? @z + Ksf (vf ? vs) = 0: (101) From equations (99)  (101), we obtain the governing equation

@us = (f )2 [( + 2 ) @ 2us ? @Tc ]: (102) s @t Ksf s @z2 @z This equation, together with its initial and boundary conditions, is solved by using the MQ method [12]. Results and Discussions.

18

The parameters used in this numerical calculations are quoted from a private communication paper of Mow et al. as follows: T = 298K R = 8:314J=mol
K cF0 = 0:15mEq=ml c = 0:15M k = 7:5M ?1 Ksf = 5.25e-14 Ns/m4 s + 2s = 0:3MPa fT = 1000kg=m3 a0 = 2:5MPa=(mEq=ml)

 =  =  = 1:0 The strain history with respect to the time t at dierent normalized height z=h is given in Fig. 3. For comparison purpose, the strain at the surface z=h = 1:0 resulted from the biphasic theory is represented by a dashed curve. It can be observed from Fig. 3 that the curves representing the strain history of the triphasic mixtures are similar to that of biphasic theory with an expectation that these strain values are lower due to the resistance from the osmotic pressure and the chemical-expansion stress. Fig. 3 indicates that the maximum strain value always occurs at the surface. Also, after the ramp displacement (t > 200s), the strain at the upper part of the tissue relaxes but still increases at the lower part until the nal equilibrated value of 10 %. Fig. 4 shows the nonlinear strain distribution with normalized height at various times. The solid curves and the dashed curves are corresponding to the compressing and relaxing stages respectively. The nal equilibrated state tends to an uniform strain distribution indicated by the curve H. For osmotic pressure and chemical-expansion stress, their history and distribution are similar to Fig. 3 and Fig. 4. The solid curves in Fig. 5 represent the time history of the pressure at dierent normalized height and the dashed curve is the total stress history. The total stress distributes uniformly throughout the height. The signi cant dierence between the pressure history and the strain history is that the pressure at all normalized height undergoes a relaxing stage, but the strain at the lower part increases to reach the equilibrated value. Fig. 6 gives the distribution of pressure with normalized height at various times. It can be seen that the pressure takes its maximum at the bottom and decreases toward surface. This pressure gradient overcomes the diusive drag and drives the uid to ow out from the surface. To further compare the dierences between the triphasic and the biphasic theories, Fig. 7 gives the history curves of the pressure and total stress near the bottom calculated using the triphasic theory (the solid curves) and the biphasic theory (the dashed curves) respectively. Curve D represents the diusive pressure pdiff , which tends to zero near the 19

equilibrated state, as calculated by

@u j ) pdiff = (Ha + 43 TC0 )( @u ? @z @z z=h

(103)

p = pdiff + posm jz=h

(104)

where Ha = s + 2s and @u=@z are the volume modulus and the strain of the solid phase respectively. If the eect of the chemical-expansion stress on the volume modulus, (4=3)TC0 , is neglected, curve E from the biphasic theory can be obtained. Curve C represents the pressure p, which is calculated by where posm jz=h is the Donnan osmotic pressure at the surface. The total stress  of the mixture, represented by the curve A, is calculated by

? = (p ? Es + TC ) jz=h :

(105)

The curve B represents the chemical-expansion stress TC jz=h but not the elastic stress ?Es jz=h of the solid phase. The total stress from the biphasic theory is indicated by the curve F . It can be observed that, according to the triphasic theory, a large part of loading in the equilibrated state is caused by the pressure, i.e. the osmotic pressure. This is dierent from the equilibrated pressure calculated from the biphasic theory, which always tends to zero. Fig. 8 shows the in unces of the salt concentration c to the strain (curves A; A1), the pressure (curves B; B1), and the total stress (curves C; C1). The solid curves and the dashed curves are corresponding to the maximum loading (t = 200s) and the equilibrium (t = 600s) state respectively. CONCLUSION Based on the triphasic mixture model developed by Lai et al. [1], all the expressions on the rate of energy, work, heat, and dissipation are given, in which the eect of the ionic concentration is expressed by the chemical energy, the eect of the chemical-expansion stress is re ected in the electrostatic energy, and the eect of the diusive resistance is included in the energy dissipation. From the generalized rst law of thermodynamics for irreversible thermodynamical system, a new set of governing equations for the triphasic mixture theory is derived in this paper. In the case of an absence of external electric eld and electric current (electroneutral solution), it has been assumed that the velocities of the free cations and of the anions 20

are equivalent, i.e. v+ = v?  vi. The ion phase can then be treated as an uniform phase for the consideration of the mechanical eects. By assuming that the ion phase ows macroscopically with respect to the water phase, i.e. vi = vw  vf , a new mechano-electrochemical biphasic mixture model is derived, which consists of (1) a porous charged solid phase including the collagen ber and proteoglycan aggregates with its xed negative charges and (2) an electroneutrally electrolytic uid (solution) phase including the interstitial water, cations and anions. This new set of governing equations for the biphasic model is directly derived from simplifying the newly derived set of equations for the triphasic mixture theory. It is also shown that, when the electrochemical eects are neglected, the basic equations of the new model can be reduced to the classical biphasic theory [3]{[5]. If only the electrostatic eects is neglected, it becomes a mechano-chemical biphasic theory similar to the model given Fung [11]. The chemical-expansion stress comes from the electrostatic eect of the xed charges. A new expression of chemical-expansion stress consisting both the in uences of deformation on the xed charge density and the electric potential of xed charge eld is presented. It is shown that the chemical-expansion stress is composed of two terms: one is related to the local tr(E) which is represented by an equivalent volume modulus 43 TC0 as given in (103) and the other, TC jz=h in (105), represents the surface chemical expansion. A one-dimensional con ned-compression problem of a charged hydrated-soft tissue is solved by using the multiquadric (MQ) method. Comparison and discussion of the results from the triphasic and biphasic theories are given. The numerical formulation and computation for the two-dimensional cases are in progress and will be reported soon. ACKNOWLEDGMENTS

This project is supported by the Research Grant Council of project No. 9040286 and the City University of Hong Kong of grant No. 7000767. The authors would also like to thank Prof. V.C. Mow and Prof. A.F.T. Mak for their invaluable comments on the rst draft of this paper.

References [1] W.M.Lai, J.S.Hou and V.C.Mow, 'A triphasic theory for the swelling and deformation behaviors of articular cartilage', ASME J. Biomech. Eng., 113, 245-258, (1991). 21

[2] W.Y.Gu, W.M.Lai and V.C.Mow, 'Transport of uid and ions through a porouspermeable charged-hydrated tissue and streaming potential data on normal bovine articular cartilage', J. Biomech., 26, 709-723, (1993). [3] V.C.Mow, S.C.Kuei, W.M.Lai and C.G.Armstrong, 'Biphasic creep and stress relation of articular cartilage in comparison: theory and experiments', ASME J. Biomech. Eng., 102, 73-83, (1980). [4] R.L.Spilker and J.K.Suh, 'Formulation and evaluation of a nite element model for the biphasic model of hydrated soft tissues', Comp. Struct., 35, 425-439, (1990). [5] J.K.Suh, R.L.Spilker and M.H.Holmes, 'A penalty nite element analysis for nonlinear mechanics of biphasic hydrated soft tissue under large deformation', Int. J. Num. Meth. Eng., 32, 1141-1439, (1991). [6] A.Maroudas, Physicochemical properties of articular cartilage, in M.A.R.Freeman (ed.), Adult Articular Cartilage, Pitman Medical, 215-290, 1979. [7] M.P.Tombs and A.R.Peacocke, The Osmotic Pressure of Biological Macromolecules, Clarendon Press, 1974. [8] V.C.Mow, A.Ratclie and A.R.Poole, 'Cartilage and diarthrodial joints as paradigms for hierarchical materials and structure', Biomaterials, 13, 67-97, (1992). [9] H.Huir, 'Proteoglycans as organisers of the intercellular matrix', Soc. Tran., 9, 489497, (1983). [10] M.A.Biot, 'New variational irreversible thermodynamics of open physical-chemical continua', Proc. of the IUTAM Symp. on Variational Methods in the Mechanics of Solid, USA, 29-39, 1978. [11] Y.C.Fung, Biomechanics: Motion, Flow, Stress and Growth, Springer-Verlag, 1990. [12] Y.C.Hon, M.W.Lu, W.M.Xue and Y.M.Zhu, 'Multiquadric method for the numerical solution of a biphasic mixture model', Appl. Math. Comp., 88, 153-175, (1997).

22

Loading Piston

z Porous Platen

c*

h Confining Chamber

Tissue Sample

o Fig. 1 Schematic of the 1-D con ned compression of a charged hydrated soft tissue equilibrated with NaCl solution

23

u(mm) 0.05

0.0

t

t0=200s

Fig. 2 Prescribed ramp displacement applied at the surface

24

0.25 Curve Normalized height(z/h) A 0.08 B 0.29 C 0.50 D 0.71 E 0.81 F 0.92 G 1.0 Bi 1.0(biphasic)

Bi

0.2

Compressive strain

G 0.15

F E D

0.1

C B A

0.05

0 0

100

200

300 Time(second)

400

Fig. 3 Strain history at dierent normalized height z=h

25

500

600

0.2 0.18 0.16

Compressive Strain

0.14 0.12

Curve A B C D E F G H

Time(s) 20 60 120 200 208 220 260 600

D E F

G H

0.1 0.08

C B

0.06 0.04

A 0.02 0 0

0.1

0.2

0.3

0.4 0.5 0.6 Normalized Height(z/h)

0.7

0.8

Fig. 4 Strain distribution with normalized height at various times

26

0.9

1

0.16 Curve A C E G

T

0.14

Pressure and Total Stress(MPa)

A 0.12

Normalized height 0.08 0.50 0.81 1.0

C 0.1 E

0.08

G

0.06

0.04

0.02

0 0

100

200

300 Time(second)

Fig. 5 Pressure and total stress history

27

400

500

600

0.14 Curve A B C D E F G H

D 0.12 C

Pressure(MPa)

0.1

E

Time(s) 20 60 120 200 208 220 260 600

F 0.08 B

0.06

G A H

0.04

0.02 0

0.1

0.2

0.3

0.4 0.5 0.6 Normalized Height(z/h)

0.7

0.8

Fig. 6 Pressure distribution with normalized height at various times

28

0.9

1

0.16 Normalized height=0.08 0.14

Pressure and Stress(MPa)

0.12

0.1 A 0.08

0.06

B D

F C

0.04 E

0.02

0 0

100

200

300 Time(second)

400

500

600

Fig. 7 Pressure and total stress history near the bottom computed by using the triphasic and biphasic theories respectively

29

0.22

Strain, Pressure(MPa), Total Stress(MPa)

0.2

A

0.18 0.16 C 0.14 0.12 A1

0.1 B 0.08

C1

0.06 B1 0.04 0.02 0

0.05

0.1

0.15 c* (M)

0.2

0.25

0.3

Fig. 8 In unces of salt concentration c to strain, pressure, and total stress

30