Fluid move in triphasic material

Collapse
X
 
  • Filter
  • Time
  • Show
Clear All
new posts
  • hipop1901
    Junior Member
    • Dec 2014
    • 27

    Fluid move in triphasic material

    Dear everyone,

    I have a problem understanding the fluid flux in triphasic material.
    In the Theory Manual:

    Equation.png
    Eq.2.126,
    k(solvent velocity- solid velocity) is related to the transient permeability, the gradient of effective pressure (first term in the parentheses), and the gradient of ion concentration. Physically, I understand effective pressure is the driven pressure and permeability determines how fast the fluid can transfer. How to understand the physical meaning of the second term in the parentheses? And under what condition, this term is important.

    Thanks,
    Bo
    Last edited by hipop1901; 07-17-2015, 05:58 PM.
  • ateshian
    Developer
    • Dec 2007
    • 1824

    #2
    Hi Bo,

    Because the effective fluid pressure includes pressure and osmolarity of the solution, the first term (gradient of effective fluid pressure) includes both permeation (fluid flux in response to a pressure gradient) and osmosis (fluid flux in response to a concentration gradient). The second term (which includes the partition coefficient and ratio of diffusivities in the mixture and in free solution) accounts for the increased resistance to solvent flux due to the frictional drag between solute and solid matrix, and gradients in electric potential. Therefore, the second term also describes electro-osmosis.

    If the solid matrix does not slow down the diffusivity of the solute (an idealized situation), then d and d0 are the same; if the solute is not charged, or if the electric potential is uniform, then the effect of the electric potential is eliminated; if we also assume that the solute solubility is 1 (i.e., if the solute can occupy all of the pore space in the solid matrix), then all these terms in the second part of the equation combine to cancel out the osmosis mechanism in the first term, so that we are left only with the permeation mechanism (Darcy's law). In summary, the second term embodies interactions that the solute has with the charged solid matrix, which influence the solvent flux.

    Best,

    Gerard
    Last edited by ateshian; 06-26-2015, 06:00 PM.

    Comment

    • hipop1901
      Junior Member
      • Dec 2014
      • 27

      #3
      Thanks!
      I still have questions about the second term:
      1. Partition coefficient: what is the sign of charged density and electrical potential (do we use the absolute value when we have negative charged density and negative electrical potential)?
      8O1S3]UEEJ7(ROQN)KQNU7N.png
      2. Electrical potential: In the free swelling process, negative electrical potential will appear in the middle of negatively charged triphasic material, same in the adding-load process. I know it may be related to the distribution of anion, cation, and negative charge. But in the steady-state process, the equivalent charge should be zero everywhere. Would you please explain how the electrical potential is calculated here?

      Bests,
      Bo

      Comment

      • ateshian
        Developer
        • Dec 2007
        • 1824

        #4
        Hi Bo,

        Fixed charge density may be positive or negative. You should not use the absolute value when the FCD is negative.

        The electric potential is obtained from the solution of the electroneutrality condition. Its sign is determined by the prevailing conditions in your analysis.

        When there are only two monovalent counter-ions in your mixture (i.e., a triphasic analysis), and assuming that the solubilities are constant, it is possible to obtain a closed-form analytical solution for the electric potential at steady state. In that case, the electroneutrality condition reduces to

        cF + cc - ca = 0

        where cF = fixed charge density, cc = cation concentration and ca = anion concentration (actual concentrations). Using the formula for the partition coefficient, and assuming that the solubilities of the anion and cation are equal to 1 (for simplicity), the relation between actual concentrations and effective concentrations may be substituted into the above equation. At steady-state equilibrium, the effective concentrations inside the mixture are uniform and equal to the effective concentrations in the external bath (cs, known from boundary conditions). This leaves the electric potential psi as the only unknown. The resulting solution is

        psi = (R T/Fc) ln[(sqrt[cF^2+4 cs^2] + cF)/(2 cs)]

        where cs = external bath salt concentration, R = universal gas constant, T = absolute temperature, Fc = Faraday's constant. This solution is known as the Donnan potential. You can verify that this solution produces negative values of psi for negative cF, and positive values for positive cF.

        Best,

        Gerard

        Comment

        • hipop1901
          Junior Member
          • Dec 2014
          • 27

          #5
          Thanks!

          1. If I understand right: in terms of partition coefficient, if we have triphasic material negatively charged, we have negative electrical potential. For example Cf=- 400mM, in 1xpbs, the electrical potential could be about -24. R = 8.314e-006, Theta =293, and Fc= 9.684e-005.
          8O1S3]UEEJ7(ROQN)KQNU7N.png
          As a result, the exponent is large negative number and partition coefficient is extremely small. Same for positively charged case. I feel this may not be right. Is it because we should use M as the unit instead of mM?

          2. In equation of Donnan Potential
          "psi = (R T/Fc) ln[(sqrt[cF^2+4 cs^2] + cF)/(2 cs)]"

          "sqrt[cF^2+4 cs^2]" is the ion concentration in material and "2 cs" is ion concentration of ion outside. We do not care about their electrical property, why do we consider the electrical property of "cF" and have it positive and negative?

          Bests,
          Bo

          Comment

          • ateshian
            Developer
            • Dec 2007
            • 1824

            #6
            Hi Bo,

            As a result, the exponent is large negative number and partition coefficient is extremely small. Same for positively charged case. I feel this may not be right. Is it because we should use M as the unit instead of mM?
            From my calculations, assuming 1xpbs is 150 mM concentration, the electric potential is -27.7 mV; the partition coefficient for the cation is 3 and that for the anion is 1/3. This means that the cation concentration inside is cc=450 while that of the anion is ca=50. This satisfies the electroneutrality condition (cF + cc - ca = -400 + 450 - 50 = 0). I don't get an "extremely small" partition coefficient. Please double check your calculations and let me know what you find.

            "sqrt[cF^2+4 cs^2]" is the ion concentration in material and "2 cs" is ion concentration of ion outside.
            sqrt[cF^2+4 cs^2] is the osmolarity inside the tissue and 2 cs is the osmolarity outside the tissue. The difference in osmolarities gives rise to the osmotic pressure.

            We do not care about their electrical property, why do we consider the electrical property of "cF" and have it positive and negative?
            The osmotic pressure arises from the difference in external and external osmolarities, irrespective of the charges. However, the main reason that osmolarities are different (other than solutes possibly having solubilities less than 1 inside the tissue) is the negative fixed-charge density inside the tissue: Because of electroneutrality, a negative FCD implies that there should be more cations than anions inside. (The opposite would be true with a positive FCD.)

            Best,

            Gerard

            Comment

            • hipop1901
              Junior Member
              • Dec 2014
              • 27

              #7
              Hi Prof. Ateshian,

              I have a question about the permeability.
              The fluid flow in my modeling was very slow so I wanted to increase the speed. Holmes-Mow description of permeability was used in the definition of material. What I did is increasing the "isotropic hydraulic permeability" K0. I do not understand why the fluid flux only increased a little (about 2 times) when I used 1000*K0. (The effective fluid pressure distributions and magnitudes in two cases are close). I expect a more linear relationship between fluid flux speed and K0.

              Thanks,
              Bo

              Comment

              • ateshian
                Developer
                • Dec 2007
                • 1824

                #8
                Hi Bo,

                The effective permeability in a multiphasic material accounts for the frictional interaction between solvent and solid (which is governed by the constitutive relation for k, such as Holmes-Mow permeability) as well as the contribution of solutes. Since solutes interact with the solvent and solid, they can slow down the transport of the solvent through the porous matrix.

                You can eliminate the contribution of the solutes completely by letting the solute diffusivity d in the mixture be the same as the solute diffusivity d0 in free solution. If you do that, the only contribution to the effective permeability will be k, in which case the fluid flux should scale with the value of k0.

                When using d=d0/2 (as in many of the triphasic examples I have posted on the forum and in the test suite for illustrative purposes), it is implied that the solutes are significantly hindered by the solid matrix. Realistically, in the case of small ions such as Na+ and Cl-, there should not be much hindrance. So, using d the same (or approximately the same) as d0 is fine in the case of these ions.

                Best,

                Gerard

                Comment

                • hipop1901
                  Junior Member
                  • Dec 2014
                  • 27

                  #9
                  Thanks, in my model d could not be set equal to d0 or it will fail during free swelling. What is the possible reason? Is it because I have a very big k0? (in my case k0 has order of 10^-13)

                  Thanks,
                  Bo

                  Comment

                  • ateshian
                    Developer
                    • Dec 2007
                    • 1824

                    #10
                    Hi Bo,

                    Would you please attache a simple example where this fails?

                    Thanks,

                    Gerard

                    Comment

                    • hipop1901
                      Junior Member
                      • Dec 2014
                      • 27

                      #11
                      Thanks, Prof. Ateshian,

                      I succeed to set d=d0 by reducing k0.

                      Comment

                      • Mediterra
                        Member
                        • Oct 2020
                        • 31

                        #12
                        Hello,

                        Following up with this topic, I'm having a hard time understanding what is the effective fluid pressure from the manual. Since the effective pressure p~ "represents that part of the fluid pressure which does not result from osmotic effects", do we consider it to be the mechanical fluid pressure, like internal pressure in arteries, or hydrostatic pressure in cartilage ?

                        Best,
                        Nicolas

                        Comment

                        • ateshian
                          Developer
                          • Dec 2007
                          • 1824

                          #13
                          Hi Nicolas,

                          The term "mechanical pressure" is not well defined but yes, if that helps, you can think of the effective fluid pressure as representing the "mechanical pressure". Others have called it the "hydraulic pressure" in contrast to the "osmotic pressure". I am not a fan of either term (mechanical vs hydraulic), which is why we use "effective pressure" in FEBio. It is important to understand that the effective fluid pressure is not the actual pressure that one would measure in the fluid, if there are osmotic effects: The actual pressure includes the osmotic and hydraulic effects. The "internal pressure in arteries" is the pressure one would measure with a pressure transducer (hydraulic + osmotic effects). There is no standard way to measure each pressure contribution separately; one can calculate each contribution, from measurements of the actual fluid pressure and actual solute concentrations, and using a suitable constitutive model that relates the osmotic pressure to the solute concentrations. Alternatively, the osmotic contribution in a fluid solution (e.g., blood) could be estimated experimentally under static conditions, by equilibrating blood against distilled water across a membrane which is impermeable to all solutes in the blood (which may be difficult to do for salt ions). The hydrostatic pressure in cartilage is similarly the combination of hydraulic and osmotic effects. For example, if the cartilage is at rest (no external loads), its interstitial fluid pressure is entirely contributed by osmotic effects. This osmotic fluid pressure is resisted by the swollen collagen matrix.

                          In the special case when the osmotic pressure contribution is zero, the effective fluid pressure is equal to the actual fluid pressure. That would occur if there are no solutes in the fluid (pure solvent, e.g., distilled water) and if the porous matrix of a tissue (such as articular cartilage) contains no fixed electrical charges.

                          I hope these clarifications help, let me know if you need a better explanation.

                          Best,

                          Gerard

                          Comment

                          • Mediterra
                            Member
                            • Oct 2020
                            • 31

                            #14
                            Hello Gerard,

                            Thank you for your answer, so if i want to describe a physiological configuration (ex: internal pressure in arteries), is the correct approach to define both pore pressure with [ p~ = p* - p_osmotic ] and surface load pressure of [ p* ] ?

                            Best,
                            Nicolas
                            Last edited by Mediterra; 12-08-2020, 02:09 AM.

                            Comment

                            • ateshian
                              Developer
                              • Dec 2007
                              • 1824

                              #15
                              Hi Nicolas,

                              If you want to prescribe a pressure p on the inner wall of an artery which is modeled as a multiphasic material, you need to do two things:

                              1) Prescribe the effective fluid pressure as p~ = p - R*T*osm (where osm = osmolarity of the fluid implicitly present inside the artery)
                              2) Prescribe a Mixture normal traction (Physics->Add Surface Load...) such that traction = -p (and Traction: Mixture)

                              Of course, this in addition to prescribing boundary conditions on the effective solute concentrations.

                              I have attached a sample file ( PressurizedArtery.feb ) that illustrates this procedure. The analysis is run under steady-state conditions.

                              Bes,

                              Gerard

                              Comment

                              Working...
                              X
                              😀
                              😂
                              🥰
                              😘
                              🤢
                              😎
                              😞
                              😡
                              👍
                              👎