Multiscale Approach to Understand the Multiphysics Phenomena in Bone Adaptation - Multiscale Computer Modeling in Biomechanics and Biomedical Engineering - page 55

Fig. 6 Donnan pressure
versus the pore size
(reproduced with permission
from Lemaire et al. [ 75 ])

justified to wonder if the new form of the Biot equation involving an electrical
coupling effect through the Maxwell tensor s [see Eq. ( 23 )] should not be

preferred.
According both to experimental studies [ 88 ] and theoretical works [ 29 ], the
most important contribution to the electro-chemical tensor s is the Donnan
osmotic pressure term p D I : An approximation of this parameter p D exists and is
proposed for a platelet geometry of the pores through the Langmuir formula which
only depends on the half inter-platelets distance d [ 53 ].
This approximation is acceptable for a pore size d larger than a few nanometers
(what is true in bone) and for high values of the surface charge density of the
interface. This surface charge density being unknown, its order of magnitude can
be estimated around 0 : 2 C.m 2 considering either the glyco-amino-glycan
surface charge [ 93 ] or the cationic absorption capacity on the hydroxyapatite
surface [ 47 ].
Moreover, the lower the salinity, the thicker the double-layer and so the
stronger the electro-chemical effects are going to be. As a result, we consider a low
salinity n b ¼ 10 4 mol : l 1 to plot in Fig. 6 the evolution of the Donnan pressure
p D versus the pore half size d : The geometry of the pore consists in face-to-face
platelets presenting the negative surface charge and separated by a distance 2d :
The calculation of the Donnan pressure results from evaluation of the double-layer
potential thanks to a custom code first developed to study clayey materials [ 80 ].
The Langmuir approximation becomes more pertinent for large pores. Never-
theless, this approximation tends to slightly overestimate the Donnan pressure.
Moreover, for typical pore sizes of the bone canaliculi (10 8 10 7 m) [ 166 ], the
Donnan pressure is one order of magnitude lower than the atmospheric pressure.
Since the typical stresses within bone lay around 0 : 1 1 MPa [ 121 , 167 ], the
electro-chemical effects included in the modified Biot constitutive can be