Results and Discussion

3-1 Code Verifications

To ensure the accuracy of the EDL potential distribution, the numerical results are also compared along with the analytical results. For the small electrostatic potential, Eq. [2-6] can be approximated by the first terms in a Taylor series. In the literature, this is call the Debye-Huckel linear approximation. This approximation is valid only for small values of electrostatic potentialψ . In this section, we will compare our numerical results with the Debye-Huckel approximation and Tuinier’s [7]

approximate solution.

3.1-1 Potential Around a Sphere

Fig 3.1 shows the numerical model for the sphere, the dimensionless electrostatic potential of the sphere surface is chosen to be constant potential. The Neumann boundary conditions are implied on the other boundaries of the domain. And in this case, the normalized radius a is chosen to be 5.

The Debye-Huckel linear approximation for sphere can defined as

( ) ( )

Where the sub “sp” refers to sphere, and a is normalized radius.

The approximate solution which proposed by Tuinier [7] is:

( ) ( )

kind of different results, are nearly all the same at smallΨ₀ =1. Cleary, our PB solver has certain accuracy. In Fig 3.3 we apply Ψ₀ =5at sphere surface. We can observe form the figures; there is a very steep decrease in the numerical results, while the Debye-Huckel approximation predicts a more gradual decay of the potential. And we also can discover Tuinier’s approximate solution slightly under predicts the numerical results.

In Fig 3.4, we discuss the convergence situation between different initial guess.

We use Debye-Huckel and Tuinier’s approximation to be initial guess, and compared with zero. Cleary, using Tuinier’s approximation to be initial guess has a good convergence situation; the Monotone sequence converges within two hundred iterations.

3.1-2 Potential Around a Cylinder

For cylindrical geometry, we do the same thing with sphere. Fig 3.5 is the numerical model for the cylinder. We also chose cylinder surface to allow Dirichlet boundary, the other surface is Neumann boundary conditions.

The Debye-Huckel linear approximation for cylindrical can defined as

( ) ( )

Where the sub “cy” refers to cylinder and K is the zeroth-order modified ₀ Bessel function of the second kind.

And the approximate solution for cylinder which proposed by Tuinier [7] is:

( ) ( )

In Fig 3.6 and 3.7 we can discover the same situation which is happened to

sphere. Debye-Huckel approximation also has wrong potential distribution at largeΨ.

3-2 Simulation of Two Interacting Identical Particles Coupled with PAMR

The problem deals with two identical colloidal particles immersed in electricity neutral electrolyte. It was studied in several works [2, 3, 5, and 6] and can server as a test. The numerical model is shown in Fig 3.8 We let segment AB to be half the separation distance which is call h, surface CD and DE are assumed the infinite of the electrolyte, surface BC represents a midplane for the problems with two particles.

In this case, the force of interaction of two particles of the radiusa=5and the separation distanceh=0.5. The constant potential Ψ on the surfaces of both ₀ particles is equal to 2.0. The Neumann boundary conditions are implied on the other boundary conditions of the numerical model.

The force of the interaction can be calculated by Eq. [2-23]. Here we will take integration over the surfaces of the particle.

In the Table 3-1 we can see the results in every refinement step; we can see the force at step 4 and step5 are almost the same. We speculate the mesh has become optimal at step 5. Fig 3.9 to Fig 3.14 is the mesh distribution for every refinement step, and Fig 3.15 is the potential distribution at final step.

It is ensured that the results shown in this article are independent of finite element mesh. The results obtained from different steps of the parallel adaptive mesh refinement are show in Table 3-1. From this table, it is clear that as we refine the mesh, the scaled of the force of the interaction obtained from the numerical solution of the Poisson-Boltzmann equation convergence toward a fixed value. After compares with literature [6], we have almost the same results, but still have inaccuracy. The previously literature’s work is shown in Table 3-2; we can see the force of the

interaction at the final step is 48.840. After compare with our result, we have 0.4 inaccuracy. At small separation distanceh=0.5 the force of the interaction is repulsion.

3-3 Two Identical Particles in a Cylindrical Pore

In this section, we will computer experiments with a geometrically confined pair of spheres concern the phenomenon of long-range electrostatic attraction between particles of like change [4]. The long-range electrostatic interaction of two colloidal particles confined in a cylindrical pore was studied in the present paper [3], we use the same values of parameters and geometry, such as the 1:1 electrolyte, the potential on surfaces of the sphere particlesΨ_s =3.0, the potential on the cylindrical pore surfaceΨ_p =5.0, the radius of the particlesa=1.185, the sphere radius to pore radius ratio 0.13. The numerical model is shown in Fig 3.8; in this case we let surface CD to be the cylindrical pore surface.

The results of the calculations are shown in Table.3-3. Positive values for the force F mean repulsion, negative mean attraction. We calculate four different _s separation distances,h=0.5, 1, 3.5, 6, 8, respectively.

Fig 3.16 and Fig 3.17 shows the calculated potential distribution for the sphere confined in a pore at separation distances h=0.5andh=6. Fig 3.18 and 3.19 shows the potential along the midplane BC ath=6. We can see the potential along this plane is almost decreasing at all distance, but after Z=3.4 the potential become increasing.

After compares with literature, we also have almost the same results, but still have 1%

inaccuracy.