In order to explain the physical reasoning, consider the binary case, in which say, copper is deposited at a surface from copper sulfate. The copper is transported to the surface to be deposited, but there can be no transport of sulfate, because it has nowhere to go. Electroneutrality requires that the concentration profiles be identical, and so in order for the flux of sulfate to be zero, the system must create a potential gradient such that migration of sulfate counters diffusion of sulfate. This gradient consequently *increases* the flux of copper to the surface as compared to diffusion alone. This is a well-established analysis, and the same logic extends to the ternary system where both non-depositing ions have zero flux.
Regarding the environment, I’m using python 3.7, fipy 3.2 converted using 2to3, and numpy 1.6.3. I apologize that this is taking up a lot of your time. ☘ > On Jun 14, 2019, at 3:39 PM, Guyer, Jonathan E. Dr. (Fed) via fipy > <[email protected]> wrote: > > 1. As before, there is no flux condition on species 3 > > 2. Unless your transport is at the speed of light, nonzero flux will not > alter Poisson's equation. That's probably a red herring, though, for now. > > More germane, the notebook at > https://urldefense.proofpoint.com/v2/url?u=http-3A__bit.ly_2wOzxfK&d=DwIGaQ&c=nE__W8dFE-shTxStwXtp0A&r=GFdsMpHIw-aduHn9jOIpyQ&m=0CNSsyiu_ttXhR5iC0w5LMRTk_0q8p-rHKbrWl-b9JI&s=BrL8P9a6glMNFn_mt5Y-XM5RutSOJZ9lSwj6ETM0FDM&e= > > <https://urldefense.proofpoint.com/v2/url?u=http-3A__bit.ly_2wOzxfK&d=DwIGaQ&c=nE__W8dFE-shTxStwXtp0A&r=GFdsMpHIw-aduHn9jOIpyQ&m=0CNSsyiu_ttXhR5iC0w5LMRTk_0q8p-rHKbrWl-b9JI&s=BrL8P9a6glMNFn_mt5Y-XM5RutSOJZ9lSwj6ETM0FDM&e=> > does not solve for me, with a singular matrix error. Replacing > C2.faceGrad.divergence with DiffusionTerm(coeff=-z1/z2, var=C1) gives a > converged solution, but again, Phi = 0. Phi = 0, C1 = C2, with grad(C1) = > grad(C2) = const is a solution to this set of equations. I still see nothing > in the conditions you have provided that would induce Phi to curve. > > I have tried both SciPy and Pysparse and I don't get the results you show for > the binary. Can you specify what environment you're running on? > >> On Jun 14, 2019, at 1:32 PM, Scott Calabrese Barton <[email protected] >> <mailto:[email protected]>> wrote: >> >> Hi, two answers: >> >> 1. Phi=0 is not a solution because it does not satisfy the flux condition on >> species 3. >> >> 2. If nonuniform charge or nonuniform permittivity are my only two choices, >> I’d go with nonuniform permittivity, due to varying concentrations. However, >> this is non-equilibrium system so I would suggest that nonzero flux can also >> give rise to potential gradients. This may be more clear in the binary >> system >> https://urldefense.proofpoint.com/v2/url?u=http-3A__bit.ly_2wOzxfK&d=DwIGaQ&c=nE__W8dFE-shTxStwXtp0A&r=GFdsMpHIw-aduHn9jOIpyQ&m=0CNSsyiu_ttXhR5iC0w5LMRTk_0q8p-rHKbrWl-b9JI&s=BrL8P9a6glMNFn_mt5Y-XM5RutSOJZ9lSwj6ETM0FDM&e= >> >> <https://urldefense.proofpoint.com/v2/url?u=http-3A__bit.ly_2wOzxfK&d=DwIGaQ&c=nE__W8dFE-shTxStwXtp0A&r=GFdsMpHIw-aduHn9jOIpyQ&m=0CNSsyiu_ttXhR5iC0w5LMRTk_0q8p-rHKbrWl-b9JI&s=BrL8P9a6glMNFn_mt5Y-XM5RutSOJZ9lSwj6ETM0FDM&e=> >> . >> >> A physical manifestation of this system is copper deposition across a >> boundary layer, say from copper sulfate with a supporting electrolyte such >> as sulfuric acid. In this case the three species might be copper, protons, >> and sulfate ions. >> >> ☘ >> >>> On Jun 14, 2019, at 11:26 AM, Guyer, Jonathan E. Dr. (Fed) via fipy >>> <[email protected]> wrote: >>> >>> Phi = 0 is a solution. Are you suggesting that this set of equations has >>> more than one solution? >>> >>> I realize you are not solving Poisson's equation. My question was to >>> consider the broader physics of the problem. Physically, non-uniform >>> electric field means either that charge is non-uniform, which is not in >>> your problem, or permittivity is non-uniform, which is not in your problem. >>> So, what's in your problem that resolves that physical inconsistency?. >>> >>>> On Jun 14, 2019, at 10:53 AM, Scott Calabrese Barton <[email protected]> wrote: >>>> >>>> To answer your questions first, the non-zero gradient in Phi counteracts >>>> the gradients in C2 and C3 such that the migration and diffusion flux >>>> cancel and overall flux is zero for those species. Regarding the >>>> curvature, you’re thinking of the Poisson equation that is only valid in >>>> this context when concentrations are constant. The variations in >>>> concentration give rise to nonlinear variations in Phi. >>>> >>>> Perhaps I should point out that we can get the numerical solution using >>>> other solvers such as solve_bvp from scipy, and an analytical solution is >>>> also available (that’s the source of the plot). So the problem is well >>>> defined. I’ll be happy to share those calculations if needed. I’m using >>>> this problem as an example to learn more about fipy. >>>> >>>>> On Jun 14, 2019, at 9:10 AM, Guyer, Jonathan E. Dr. (Fed) via fipy >>>>> <[email protected]> wrote: >>>>> >>>>> I had the same initial thought, but you're not adding new information by >>>>> doing that. You can see this by looking at the matrix stencil that >>>>> results from these equations: >>>>> >>>>> X X C1 >>>>> X C2 >>>>> XX C3 = 0 >>>>> XXX Phi >>>>> >>>>> That bottom right sector of the matrix is empty and so the matrix solvers >>>>> cannot find a solution because you're now trying to solve for four >>>>> unknowns with what are really only three equations. >>>>> >>>>> Things to consider: >>>>> >>>>> In the solution you posted from West, Phi does not have zero gradient at >>>>> either end of the domain. Why? What boundary condition is driving that? >>>>> >>>>> Phi is curved, which implies that there is net charge in the system, but >>>>> you assert zero charge throughout. How does West reconcile that >>>>> inconsistency? >>>>> >>>>>> On Jun 13, 2019, at 11:25 AM, Scott Calabrese Barton <[email protected]> >>>>>> wrote: >>>>>> >>>>>> That’s true, so we can make C3 explicit (full code at >>>>>> https://urldefense.proofpoint.com/v2/url?u=http-3A__bit.ly_2F7UQgR-3D&d=DwIGaQ&c=nE__W8dFE-shTxStwXtp0A&r=GFdsMpHIw-aduHn9jOIpyQ&m=NXf47PawWEovDQCuiqrE25DZuppCDLEVN9L9lPfcl3c&s=3mcnknSVK1Ij82U2PK6itcevIOeNgXVMqLhUYYihMqc&e= >>>>>> ): >>>>>> >>>>>> # Variables >>>>>> C1= CellVariable(mesh = mesh, value = 1.0) >>>>>> C2= CellVariable(mesh = mesh, value = 4.0/3.0) >>>>>> Phi = CellVariable(mesh = mesh, value = 0) >>>>>> >>>>>> C30 = -1/z3 * (z1*1.0 + z2*C20) # Electroneutrality >>>>>> C3= CellVariable(mesh = mesh, value = C30) >>>>>> >>>>>> # Boundary Conditions >>>>>> C1.constrain(1.0, mesh.facesRight) >>>>>> C2.constrain(C20, mesh.facesRight) >>>>>> C3.constrain(C30, mesh.facesRight) >>>>>> Phi.constrain(-0.2, mesh.facesRight) >>>>>> C1.constrain(0.0, mesh.facesLeft) >>>>>> >>>>>> # Governing Equations >>>>>> Eq1 = + DiffusionTerm(coeff=z1*C1, var=Phi) + DiffusionTerm(coeff=1.0, >>>>>> var=C1) >>>>>> Eq2 = + DiffusionTerm(coeff=z2*C2, var=Phi) + DiffusionTerm(coeff=1.0, >>>>>> var=C2) >>>>>> Eq3 = + DiffusionTerm(coeff=z3*C3, var=Phi) + DiffusionTerm(coeff=1.0, >>>>>> var=C3) >>>>>> >>>>>> Eq4 = ImplicitSourceTerm(coeff=z1, var=C1) + \ >>>>>> ImplicitSourceTerm(coeff=z2, var=C2) + \ >>>>>> ImplicitSourceTerm(coeff=z3, var=C3) >>>>>> >>>>>> Eqns = Eq1 & Eq2 & Eq3 & Eq4 >>>>>> >>>>>> >>>>>> However, this set of equations does not converge. I’m fairly certain >>>>>> that I am mismanaging the boundary conditions. >>>>>> >>>>>> >>>>>>> On Jun 12, 2019, at 5:02 PM, Guyer, Jonathan E. Dr. (Fed) via fipy >>>>>>> <[email protected]> wrote: >>>>>>> >>>>>>> There's no zero-flux BC on C3 because there's no such thing as C3 as >>>>>>> far as FiPy is concerned. The boundary conditions on C3 are entirely >>>>>>> determined by the boundary conditions on C1 and C2 (Dirichlet on the >>>>>>> right and Robin on the left). The solution obtained satisfies the >>>>>>> equations and boundary conditions given to FiPy. >>>>>>> >>>>>>>> On Jun 12, 2019, at 2:19 PM, Scott Calabrese Barton <[email protected]> >>>>>>>> wrote: >>>>>>>> >>>>>>>> Jon, thanks for catching that typo. I wish it were the problem! >>>>>>>> >>>>>>>> The solution with grad phi=0 does not satisfy the zero-flux boundary >>>>>>>> condition on species 3. In the absence of a potential gradient, that >>>>>>>> species would need to also have zero gradient (like species 2 in the >>>>>>>> current solution). >>>>>>>> >>>>>>>> So the question becomes how to enforce that zero-flux BC. Any >>>>>>>> suggestions? I could drop the conservation equation and replace it >>>>>>>> with something that enforces zero-flux everywhere, but I’d prefer to >>>>>>>> use a boundary condition as that would be more applicable in higher >>>>>>>> dimensions. >>>>>>>> >>>>>>>> ☘ >>>>>>>> >>>>>>>> >>>>>>>>> On Jun 12, 2019, at 1:36 PM, Guyer, Jonathan E. Dr. (Fed) via fipy >>>>>>>>> <[email protected]> wrote: >>>>>>>>> >>>>>>>>> Scott - >>>>>>>>> >>>>>>>>> Eq2 appears to have a typo: >>>>>>>>> >>>>>>>>> -Eq2 = DiffusionTerm(coeff=z2*C1, var=Phi) + DiffusionTerm(coeff=1.0, >>>>>>>>> var=C2) >>>>>>>>> +Eq2 = DiffusionTerm(coeff=z2*C2, var=Phi) + DiffusionTerm(coeff=1.0, >>>>>>>>> var=C2) >>>>>>>>> >>>>>>>>> Eq3 has (much) better coupling if written like this: >>>>>>>>> >>>>>>>>> -Eq3 = DiffusionTerm(coeff=z3*C3, var=Phi) + (C3.faceGrad).divergence >>>>>>>>> +Eq3 = DiffusionTerm(coeff=z3*C3, var=Phi) - >>>>>>>>> DiffusionTerm(coeff=-z1/z3, var=C1) - DiffusionTerm(coeff=-z2/z3, >>>>>>>>> var=C2) >>>>>>>>> >>>>>>>>> [Note that this way you don't need to evaluate .faceGrad at the >>>>>>>>> boundaries, so your issue #650 doesn't come up. We don't upwind at >>>>>>>>> the boundaries because it's complicated and expensive to do right and >>>>>>>>> it's unusual to care.] >>>>>>>>> >>>>>>>>> With these changes, the solution converges in one sweep, but does not >>>>>>>>> agree with the Analytical Solution you give from West. >>>>>>>>> >>>>>>>>> I don't understand a couple of things though. As posed, what governs >>>>>>>>> \Phi? I get \Phi = 0 and see no reason why it would be otherwise. >>>>>>>>> Typically, I would expect Poisson's equation to be obeyed. Since you >>>>>>>>> assert charge electroneutrality, this reduces to Laplace's equation, >>>>>>>>> but in that case, the slope of \Phi should be constant (unless the >>>>>>>>> permittivity is non-uniform, but you don't have a permittivity). In >>>>>>>>> the Analytical plot, \Phi has definite curvature. >>>>>>>>> >>>>>>>>> My guess is you don't need Eq3 (it'll get taken care of >>>>>>>>> automatically) and you need some form of Poisson's equation. >>>>>>>>> >>>>>>>>> - Jon >>>>>>>>> >>>>>>>>>> On Jun 11, 2019, at 4:48 PM, Scott Calabrese Barton <[email protected]> >>>>>>>>>> wrote: >>>>>>>>>> >>>>>>>>>> I’ve had some trouble using fipy for the 1D, steady-state ternary >>>>>>>>>> electrolyte problem, which involves coupled, nonlinear migration and >>>>>>>>>> diffusion. >>>>>>>>>> >>>>>>>>>> A text version of the model is listed below. A more complete >>>>>>>>>> development is at >>>>>>>>>> https://urldefense.proofpoint.com/v2/url?u=http-3A__bit.ly_2wJDXob-3D-3D&d=DwIGaQ&c=nE__W8dFE-shTxStwXtp0A&r=GFdsMpHIw-aduHn9jOIpyQ&m=NXf47PawWEovDQCuiqrE25DZuppCDLEVN9L9lPfcl3c&s=7BJyGej3bHZQwW3KYHek8AbNpqq32E63p3rH-Vvq9UM&e= >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> from fipy import * >>>>>>>>>> >>>>>>>>>> # Parameters >>>>>>>>>> nx=50 >>>>>>>>>> C20=4.0/3.0 >>>>>>>>>> z1, z2, z3= 2.0, -2.0, 1.0 >>>>>>>>>> >>>>>>>>>> # Mesh >>>>>>>>>> mesh = Grid1D(nx=nx, dx=1.0/nx) >>>>>>>>>> >>>>>>>>>> # Variables >>>>>>>>>> C1= CellVariable(mesh = mesh, value = 1.0) >>>>>>>>>> C2= CellVariable(mesh = mesh, value = 4.0/3.0) >>>>>>>>>> Phi = CellVariable(mesh = mesh, value = -0.5) >>>>>>>>>> >>>>>>>>>> # Electroneutrality >>>>>>>>>> C3 = -1/z3 * (z1*C1 + z2*C2) >>>>>>>>>> >>>>>>>>>> # Boundary Conditions >>>>>>>>>> C1.constrain(1.0, mesh.facesRight) >>>>>>>>>> C1.constrain(0.0, mesh.facesLeft) >>>>>>>>>> C2.constrain(C20, mesh.facesRight) >>>>>>>>>> Phi.constrain(0, mesh.facesRight) >>>>>>>>>> >>>>>>>>>> # Governing Equations >>>>>>>>>> Eq1 = DiffusionTerm(coeff=z1*C1, var=Phi) + DiffusionTerm(coeff=1.0, >>>>>>>>>> var=C1) >>>>>>>>>> Eq2 = DiffusionTerm(coeff=z2*C1, var=Phi) + DiffusionTerm(coeff=1.0, >>>>>>>>>> var=C2) >>>>>>>>>> Eq3 = DiffusionTerm(coeff=z3*C3, var=Phi) + (C3.faceGrad).divergence >>>>>>>>>> >>>>>>>>>> Eqns = Eq1 & Eq2 & Eq3 >>>>>>>>>> >>>>>>>>>> # Solution >>>>>>>>>> res = 1e+10 >>>>>>>>>> restol= 1e-3 >>>>>>>>>> >>>>>>>>>> while res > restol: >>>>>>>>>> res = Eqns.sweep() >>>>>>>>>> print(res) >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> However, this system does not converge. I suspect the issue is with >>>>>>>>>> boundary conditions. I’ve tried zeroing out the diffusion >>>>>>>>>> coefficients at the boundary, but that does not seem to make a >>>>>>>>>> difference here. I am able to use this approach for the analogous >>>>>>>>>> binary problem: >>>>>>>>>> https://urldefense.proofpoint.com/v2/url?u=http-3A__bit.ly_2wOzxfK-3D-3D&d=DwIGaQ&c=nE__W8dFE-shTxStwXtp0A&r=GFdsMpHIw-aduHn9jOIpyQ&m=NXf47PawWEovDQCuiqrE25DZuppCDLEVN9L9lPfcl3c&s=m59IS6E5x3ikFID0-LV2HY3XwhWMEDBSf7p0NOFfTSI&e= >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> Any insights or suggestions would be welcome. >>>>>>>>>> >>>>>>>>>> Cheers, >>>>>>>>>> Scott☘ >>>>>>>>>> >>>>>>>>>> >>>>>>>>>> _______________________________________________ >>>>>>>>>> fipy mailing list >>>>>>>>>> [email protected] >>>>>>>>>> https://urldefense.proofpoint.com/v2/url?u=http-3A__www.ctcms.nist.gov_fipy-3D-3D&d=DwIGaQ&c=nE__W8dFE-shTxStwXtp0A&r=GFdsMpHIw-aduHn9jOIpyQ&m=NXf47PawWEovDQCuiqrE25DZuppCDLEVN9L9lPfcl3c&s=ReSWGMkuYcc6b3YlxVB_e86OwfNxqVxp12I2NbAXyyM&e= >>>>>>>>>> >>>>>>>>>> [ NIST internal ONLY: >>>>>>>>>> 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