(217e) Diffusioosmosis of an Electrolyte Solution along a Plane Wall

Diffusioosmosis of an electrolyte solution is the electrokinetic flow involving a tangential concentration gradient of the electrolyte that interacts with a charged wall. Electrolyte solutions with a concentration gradient of order 1 M/cm along solid surfaces with a zeta potential of order kT/e (~25 mV; e is the charge of a proton, k is the Boltzmann constant, and T is the absolute temperature) can flow by diffusioosmosis at a velocity of order μm/s.  The fluid motion caused by diffusioosmosis was analytically examined for the flow near a plane wall [1-6].  In these studies, however, the effect of lateral distributions of the counterions and co-ions on the local electric field induced by the imposed electrolyte concentration gradient in the tangential direction inside the electric double layer or the effect of the ionic convection on it caused by the diffusioosmotic flow was neglected.

In this work we present a comprehensive analysis of the diffusioosmosis of an electrolyte solution with a constant prescribed concentration gradient along a charged plane wall.  No assumption is made concerning the magnitude of the electric potential or the thickness of the double layer, and both the lateral distribution of the induced tangential electric field and the effect of the electrolyte convection on it are allowed.  Semi-analytical results for the fluid velocity profile and the bulk-phase diffusioosmotic velocity are obtained for various cases.

2. Analysis

We consider the diffusioosmotic flow of an infinitely thick solution of a symmetric electrolyte of valence Z tangential to a uniformly charged plane wall at the steady state.  The applied electrolyte concentration gradient  is a constant along the tangential (z) direction, where  is the linear concentration (number density) distribution of the electrolyte in the bulk solution phase far from the wall (with ).  It is assumed that  is only slightly nonuniform such that the variation of the electrostatic potential and ionic concentrations in the double layer adjacent to the wall with the tangential position can be neglected in comparison with their corresponding quantities at  (the midpoint along the plane wall).

The electrostatic potential distribution in the electrolyte solution near the charged plane wall, , is the well-known Gouy-Chapman result [1],

,                                                                                       (1)

where , , ,  is the zeta potential of the wall, and , where  is the relative permittivity of the electrolyte solution and  is the permittivity of a vacuum.

        The local concentrations  and  of the cations and anions, respectively, in the fluid undergoing diffusioosmosis along the plane wall are not uniform in both tangential  and normal  directions, and their gradients in the tangential direction can give rise to a ?diffusion current? distribution.  To prevent a continuous separation of the counterions and co-ions, an electric field distribution along the tangential direction arises spontaneously in the electrolyte solution to produce another electric current distribution which exactly balances the diffusion current.  This induced electric field generates an electroosmotic flow of the fluid parallel to the plane wall, in addition to the chemiosmotic flow caused by the electrolyte gradient directly.  Both the chemiosmotic and the electroosmotic flows also generate an electric current distribution by the ionic convection, and alternately, this secondary ?convection current? again needs to be balanced by the electric current contributed from the induced electric field.

The total flux of either ionic species can be expressed as the general form

                ,                                                              (2)

where  is the fluid velocity relative to the plane wall in the direction of decreasing electrolyte concentration (i.e.,  is the unit vector in the direction of ),  and  are the diffusion coefficients of the cations and anions, respectively,  is the macroscopic electric field induced by the concentration gradient of the electrolyte.  To have no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the cations and anions, one must require that .  Applying this constraint to Eq. (2), we obtain

                ,                       (3)

where

,                                                                          (4)

which is a characteristic value of the diffusioosmotic velocity,

                ,                                                                                                       (5)

                ,                                                                           (6)

and  is the fluid viscosity.  Evidently, the induced electric filed  given by Eq. (3) in a self-consistent way depends on the local electrostatic potential  and fluid velocity .

        For the steady diffusioosmotic flow along a plane wall, the momentum balance on the electrolyte solution in the  direction gives

    ,                                                        (7)

where the pressure distribution

    .                                                                         (8)

and  is the pressure far away from the wall, which is a constant in the absence of the applied pressure gradient. 

Substituting the Boltzmann distributions of  and  and the pressure profile of Eq. (8) into Eq. (7) and then performing the integration with respect to y twice subject to appropriate boundary conditions, we obtain

.                           (9)

After the substitution of Eq. (9) for u and Eq. (1) for  into Eq. (3), the induced electric field E can be numerically solved as a function of the parameters , , , and Pe.  With the known results of  and E, the diffusioosmotic velocity distribution of the electrolyte solution can be determined from Eq. (9) with the numerical integrations.  The bulk-phase diffusioosmotic velocity  of the electrolyte solution at a large distance from the plane wall as a function of the relevant parameters can be obtained from the distribution u by taking .

3. Results and discussion

The results of the normalized diffusioosmotic velocity distribution are plotted in Figure 1.  For fixed values of the parameters , Pe, and , the value of  approaches a constant (equal to ) as .  When Pe=0 and the product of  and  is negative,  is also negative, meaning that the diffusioosmotic flow is in the direction of increasing electrolyte concentration.  The magnitude of  increases monotonically with an increase in the electrokinetic distance  from the wall, with an increase in , and with an increase in , for an otherwise specified condition.  When Pe=0 and  is positive, the direction of the diffusioosmotic flow depends on the combination of parameters , , and .  When the value of Pe is finite, the dependence of  on  is similar to that for the case of Pe=0 if the value of  is small, but  can be positive for any given value of  if the value of  is large.  In general, the value of  increases monotonically with an increase in the value of Pe for specified values of , , and  except for the situation that the value of  is close to unity.

Maps showing the direction of the normalized diffusioosmotic velocity  of the electrolyte solution far away from the plane wall for various values of the parameters Pe, , and  are drawn in Figure 2.  When Pe=0 and the product of  and  is negative (inside the second and fourth quadrants in Figure 6a),  is negative and the electrolyte solution flows toward higher concentration.  When Pe=0 and the product of  and  is positive (inside the first and third quadrants in Figure 6a), the bulk-phase diffusioosmotic velocity may reverse its direction from against the concentration gradient to along with it as  increases not much from zero for the case of a small magnitude of , or as  decreases from 1 to 0 for a given value of .  When the value of Pe is finite, as indicated in Fig. 2b, the diffusioosmotic flow of the electrolyte solution in many cases (such as that with a negative value of  and a sufficiently large magnitude of ) may reverse its direction from toward higher concentration to toward lower concentration, due to the effect of the electrolyte convection. References

[1]       S.S. Dukhin, B.V. Derjaguin, in: E. Matijevic (Ed.), Surface and Colloid Science, vol. 7, Wiley, New York, 1974.

[2]       D.C. Prieve, J.L. Anderson, J.P. Ebel, M.E. Lowell, J. Fluid Mech.148 (1984)247.

[3]       J.L. Anderson,Annu. Rev. Fluid Mech. 21(1989) 61.

[4]       H.J. Keh, S.B. Chen, Langmuir 9(1993) 1142.

[5]       Y. Pawar, Y.E. Solomentsev, J.L. Anderson, J. Colloid Interface Sci.155(1993) 488.

[6]       H.J. Keh, H.C. Ma, Colloids Surfaces A 233 (2004) 87.

Figure 1. The normalized diffusioosmotic velocity  versus  for various values of  with Pe=1 (solid curves) and Pe=0 (dashed curves): (a) ; (b) .

Figure 2. Maps showing the direction of the bulk-phase diffusioosmotic velocity : (a) Pe=0; (b) Pe=1.