Transference Number Approaching Unity in Nanocomposite

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Transference Number Approaching Unity in Nanocomposite Electrolytes

2006 Vol. 6, No. 12 2973-2976

Jacob Jorne* Department of Chemical Engineering, UniVersity of Rochester, Rochester, New York 14627 Received September 15, 2006; Revised Manuscript Received October 11, 2006

ABSTRACT In a nanocomposite electrolyte, which is made of a dielectric solid and an electrolyte, the diffuse double layer can retain counter-ions and reject co-ions. The axial external field can then interact to produce hydrodynamic flow. If the radius of the nanocapillaries is on the same order as the Debye length of the electrolyte, the transference number of an ion can approach unity, eliminating concentration gradients. Nanocomposite electrolytes are therefore suggested for battery and fuel cell applications where the transference number of the transported ion approaches unity, thus eliminating concentration gradients and reducing energy losses.

Introduction. One of the fundamental problems in electrochemistry is the fact that in most electrolytes, the transference number of a specific ion is always less than unity. This is a result of the fact that all ions participate in the conductance process as electroneutrality must be maintained everywhere in the electrolyte. Consequently, concentration gradients are established within the electrolyte as diffusion supplements migration in the transporting of the reactive ions to and from the electrodes. Thus, concentration overpotential is a direct result of the fact that the transference number of an ion is less than unity. In batteries, for example, large concentration overpotentials lead to power losses, decomposition of the electrolyte, dendritic growth, and other adverse effects. It is therefore highly desirable to increase the transference number of the reactive ion as close as possible to unity. In certain solid electrolytes and membranes, the transference number is unity because the counterion is immobilized as it is attached to the solid matrix. The only region in the liquid electrolyte where local electroneutrality is not maintained is the double layer, where the charge on the solid is counterbalanced by the ions in the double layer. For example, if the surface is negatively charged, the cations in the electrolyte are attracted while the anions are repelled and the transference number of the cations could approach unity in the diffuse double layer. The same can be observed when the surface is positively charged and the diffuse double layer contains mostly anions and their transference number there could approach unity. However, the diffuse double layer is very narrow, on the order of the Debye length, and therefore contributes very little to the overall conductance. Further, in a porous medium, where the size of the pores is similar or smaller than the Debye * E-mail: [email protected] 10.1021/nl062182m CCC: $33.50 Published on Web 11/01/2006

© 2006 American Chemical Society

length, most of the electrolyte in the pores is a part of the diffuse double layer where electroneutrality is not maintained and the transference number of the counterion can approach unity. Transference Number in Nanocomposite Electrolyte. Let us consider a porous dielectric material filled with an electrolyte. The capillaries have a pore radius r0, which is on the order of the Debye length. The Debye length depends on the ionic strength of the electrolyte and varies from about 10 nm for 1 mM to 0.3 nm for 1M solutions. A constant electric field is imposed in the axial direction, however, because of the charge on the solid, an electric field also exists in the radial direction. Newman1 treats the electrokinetic phenomena that occur when the diffuse double layer and the external field interact to produce hydrodynamic flow in the capillaries. This process is known as electro-osmosis. The electric potential Φ in the electrolyte is given by the Poisson’s equation: Fe 1 d dΦ r )r dr dr 

( )


where Fe is the charge density, r is the spherical coordinate, and  is the dielectric constant, and where we assume that the axial electric field Ez is constant. In the absence of a pressure gradient across the pores, the momentum equation in the z-direction is given by:

( )

µ d dVz r + EzFe ) 0 r dr dr


where Vz is the local velocity in the axial direction and µ is

() ()()

the viscosity of the electrolyte. Combining the two equations and following double integration, subjected to the boundary conditions: r ) r0, Vz ) 0, Φ - Φ0 ) 0; r ) 0, Vz ) finite, Φ ) finite, gives: V z ) Ez

 (Φ - Φr)0) µ


r0 q2r0 I2 λ 〈Vz〉 ) E µ r0 r0 z I λ 1 λ

where I2 is the modified Bessel function of the first kind of order two. In the absence of axial diffusion, the ion flux Ni is the sum of the electric field and the flow:

The radial distribution of the ion concentration in the capillary ci is given by the Boltzmann distribution and the Gouy-Chapman model for the diffused double layer:


ci ) ci0 exp -

ziF (Φ - Φr)0 RT



where ci0 is the concentration at the center line of the capillary and Φr)0 is the potential. zi is the charge number and F is the Faraday constant. The Gouy-Chapman theory for the diffuse double layer assumes a point ion approach and neglects the limitation of the distance of closest approach of ions to the surface. These limitations are expected to be important in nanogeometries. For small potentials, using the Debye-Huckel approximation, the exponent in eq 4 can be approximated by a linear Taylor’s series expansion:



ziF(Φ - Φr)0) ci = ci0 1 RT

Φ - Φr)0 )

q2λ r 1 - I0 r0 λ I1 λ



( )]


where I0 and I1 are the modified Bessel functions of the first kind of the order zero and one, respectively. λ is the Debye length: λ)




∑ zi2ci0



[ ( ) ( )]

r0 r I0 - I0 Ez λ λ


∑ ziNi

iz ) F


The total current due to ion i is given by: Ii ) 2π


) 2π



r(F2Ezzi2uici + FVzzici)r dr


r F2Ezzi2uici0 1 -



Fzici0 1 -



ziF(Φ - Φr)0) RT

{ [ [ ( )] [

) 2πEzci0

Fzi 1 -




ziF(Φ - Φr)0) + RT λq2 r0 µI1 λ


[ ( ) ( )] I0


r0 r - I0 Ez dr λ λ


ziF λq2 r 1 - I0 + RT r0 λ I1 λ λq2 r0 r r 1 - I0 I - I0 r dr λ r0 0 λ λ µI1 λ (12)

F2zi2ui 1 -

ziF λq2 RT r0 I1 λ

( )]

() ( )] [ ( ) ( )] ()


The total current density is the sum of the ionic currents: iz ) F

∑ ziNi ) F2Ez ∑ zi2uici + FVx ∑ zici z

) κEz + VzFe



where κ ) F2 ∑ zi2uici is the conductivity and Fe ) ∑ zici is the charge density. The total current I is given by integration over the area of the capillary (eq 1): I ) 2π


) Ez (8)

The average solvent velocity in the pore 〈Vz〉 is evaluated by integrating the local Vz over the pore cross section:


where ui is the ionic mobility and the local current density is:


The dielectric constant of the solvent  is known to be smaller in confined geometries where strong electric fields order the solvent molecules. It is therefore expected that the Debye length would be smaller inside nanogeometries. The local velocity is then: λq2 Vz ) r0 µI1 λ

Niz ) ziuiFciEz + Vci


The potential distribution, subjected to a constant surface charge density q2, is given by (eq 1):



riz dr ) Ez



rκ dr -


rκ dr +

Ez2 µ





rVzFe dr


(Φ - Φr)r0)

d dΦ r dr (14) dr dr

( )

and ) Ez



rκ dr + Ez

2 µ




dr (dΦ dr ) 2

Nano Lett., Vol. 6, No. 12, 2006

{ [

()() ( ( ))

r0 r0 2 I0 I q 2 λ 2 λ I ) πr02Ez κavg + 1µ r0 2 I1 λ


Table 1. Ratios of Bessel’s Functions


where κavg is the average conductivity: ∫r00 rκ dr ) r02/2 κavg. In the present case, the transference number represents the fraction of the current carried by ion i because there are no concentration variations. The fraction of the current carried by ion I is given by:




()() () ()() ()

r0 r0 2 I0 I q 2 λ 2 λ 1F2ziuiciavg + µ r 2 0 I 1 Ii λ ti ) ) I r0 r0 I0 I q22 λ 2 λ κavg + 1µ r0 I12 λ



When the term q22/µ [1 - I0(r0/λ)I2(r0/λ)/I12(r0/λ)] is dominant, then ti approaches unity. Table 1 presents ratios of the Bessel functions as a function of r0/λ. The ratio I2/I1 I0/I1 is significantly smaller than unity when r0/λ varies between 0 ∼ 1, therefore the radius of the capillaries must be of the same order as the Debye length λ in order for the electrokinetic effect to be significant and the transference number to approach unity. However, when r0/λ . 1, the transference number assumes its usual definition:

ti )

F2ziuiciavg κavg


For r0/λ ∼ 1, the ratio of the Bessel’s functions remains about 0.5, as can be seen from Table 1. Typically, the surface charge density is q2 ) 0.5 µC/cm2 and the viscosity is µ ) 10-2 g/cm s. For typical electrolytes, κ ∼ 10-2 ohm-1 cm-1 and the fraction of the current carried by ion i approaches unity, ti ∼ 1.0. This approach to unity occurs because, under such conditions, the pores contain mostly ion i, while the counter ions are repelled by the surface charge density q2, and the transport is assisted by the electro-osmosis-driven hydrodynamic flow. Application to Batteries and Fuel Cells. On the basis of the present theoretical analysis, the incorporation of nanocomposite electrolytes in batteries and fuel cells can eliminate concentration variations and mass transfer limitations. These composite electrolytes can be in the form of a mixture of solid polymer electrolyte mixed with dielectric materials such as alumina or porous silica. This incorporation of nanocomposite electrolytes can also be applied to liquid electrolyte, where the solvent fills the void volume of the nanoporous dielectric matrix. The typical diameter of the pores should be in the nanorange and comparable to the Debye length of the electrolyte, while, simultaneously, the volume fraction of the solid should be quite low as to not significantly reduce the effective conductivity of the composite. Nano Lett., Vol. 6, No. 12, 2006

r0 λ

I 2 I0 I1 I1

0 0.1 0.2 1.0 2.0 5.0 10.0 ∞

0 0.500 0.5015 0.538 0.620 0.805 0.871 1.0

Recently, Liu et al.2 enhanced the apparent proton conductivity inside a nanochannel by orders of magnitude due to the electric double layer overlap. Their results support the theory presented here. Furthermore, they developed a new class of proton exchange membrane for micro fuel cell applications. Nan et al.3 also significantly enhanced the ionic conductivity of polymer electrolyte containing nanocomposite SiO2 particles. Rechargeable lithium ion batteries are the leading high power batteries. The currently available lithium ion batteries include those based on both solid polymer electrolytes (SPE) and on aprotic solvents. The polymer electrolytes have a typical thickness of about 50-100 µ and consist of solid polymer polyethylene oxide (PEO) and other polymers complexed with lithium salts and saturated with aprotic solvents. These electrolytes suffer from low conductivities and a low transference number for the lithium ion (t < 0.5), which results in concentration gradients and energy and power losses. Peled et al.4,5 and Nagasubramania et al.6 found that composite polymer electrolytes, in which the polymer is mixed with alumina Al2O3 and LiI, present higher conductivities and remarkable high transference number, approaching unity. On the basis of these results, Peled et al.4,5 constructed a lithium composite electrolyte battery with superior performance. Nishizawa et al.7 observed that membranes containing metal nanotubes show selective ion transport analogous to that observed in ion-exchange polymers. They found that the concentration cells separated by these membranes exhibit a transference number of unity, and the selectivity can be controlled potentiostatically. The fabricated nanocomposite electrolytes are expected to present superior performance of mechanical strength, cracking, brittleness, internal shorts, and dendritic growth. The electrolyte, either solid or liquid, is constrained within and supported by the inert dielectric matrix. The charge density q2 is the result of the surface adsorption of counterions on the dielectric surface. It is therefore important that strong specific adsorption is present. The active electrolyte, whether liquid, solid, polymer, or gel, is introduced into the porous structure by either capillary action or by deposition. It is expected that dendritic growth will be prevented by the close-to-unity transference number. Nanoporous membranes are commercially available. Alumite membranes, for example, are produced by the anodization of aluminum. Matrices of nanohole arrays can be fabricated by replication processes. Masuda and Fukuda,8 among others, fabricated highly 2975

ordered nanohole arrays by the two-step replication of honeycomb-structured anodized porous alumina. The diameter of the nanoholes was approximately 70 nm. This manufacturing process can well serve the fabrication of nanocomposite-constrained electrolytes. References (1) Newman, J. S.; Thomas-Alyea, K. E. Electrochemical Systems, 3rd ed.; Wiley Interscience: New York, 2004. (2) Liu, S.; Pu, Q.; Gao, L.; Korzeniewski, C.; Matzke, C. Nano Lett. 2005, 5, 1389-1393. (3) Nan, C.-W.; Fan, L.; Lin, Y.; Cai, Q. Phys. ReV. Lett. 2003, 91, 266104-1-266104-4.


(4) Peled, E.; Golodnitsky, D.; Lang, J.; Lavi, Y. Proceedings of the 184th Meeting of the Electrochemical Society, New Orleans, LA, October 13, 1993. (5) Peled, E.; Golodnitsky, D.; Menachem, C.; Ardel, G.; Lavi, Y. Proceedings of the 184th Meeting of the Electrochemical Society, New Orleans, LA, October 13, 1993. (6) Nagasubramanian, G.; Attia, A. I.; Halpert, G. Solid-State Ionics 1993, 67, 51. (7) Nishizawa, M.; Menon, V. P.; Martin, C. R. Science 1995, 268, 700702. (8) Masuda, H.; Fukada, K. Science 1995, 268, 1466.


Nano Lett., Vol. 6, No. 12, 2006