Theory of ionic conductivity with morphological control in polymers

Abstract

We present a general theory of ionic conductivity in polymeric materials consisting of percolated ionic pathways. Identifying two key length scales corresponding to inter-path permeation distance ξ and one-dimensional hopping conduction path length mλ, we have derived closed-form formulas in terms of the energy U required to unbind a conductive ion from its bound state and the partition ratio ξ/mλ between the three-dimensional permeation and one-dimensional hopping pathways. The results provide design strategies to significantly enhance ionic conductivity in single-ion conductors. For large barriers to dissociate an ion, corrections to the Arrhenius law are presented. The predicted dependence of ionic conductivity on the unbinding time is in agreement with results in the literature based on simulations and experiments. This theory is generally applicable to conductive systems where the two mechanisms of permeation and hopping occur concurrently.

The subject of ionic conductivity in polymeric materials with heterogeneous structures is of intense current interest, primarily due to the societal need for better battery alternatives [1–32]. Significant progress has recently been made towards formulating polymeric materials with enhanced ionic conductivities and at the same time not compromising on their mechanical stability. The procedures that have been implemented in this endeavor are primarily experiments and simulations [1–32]. Generally speaking, the investigated systems include solid polymer electrolytes such as salt-doped poly(ethylene oxide), polymeric single-ion conductors, polymerized ionic liquids, and polyelectrolyte solutions in nano-capillaries [1–32]. The combined approach of precise synthesis of ion-containing polymers, characterization of their assembled structures using a variety of experimental techniques, and molecular modeling has revealed exquisite details on several specific systems and their consequence on the temperature dependence of ionic conductivity. Nevertheless, it is desirable to develop fundamental relations between the heterogeneous structures and the consequent ionic conductivity that are universal instead of treating each system as unique. Such relations would enable design principles to achieve desirable ionic conductivities by tuning the morphology of heterogeneous polymeric materials. With this goal in mind, we present in this Letter a theory for ionic conductivity in polymeric materials with heterogeneous structures.

For illustrative purposes, let us consider three scenarios of general context (Fig.1). The first is pertinent to ion transport in nanoporous media (Fig.1a), where the surface of interconnected pores carry immobile charges and the interior of the pores permit conduction of oppositely charged counterions. In general, the counterions are bound to the charged groups on the interface and the extent of binding depends on the specificity of the ionic species and local dielectric environment. The movement of bound counterions under an external electric field can occur either by hopping (sliding) to its neighboring charged group on the interface or by permeation (conduction) through the interior medium of the pores. These two pathways are indicated by the solid and broken arrows in Fig.1a. In addition to the rigid inorganic porous materials, hydrated Nafion-like organic materials (where interconnected network of hydrophilic domains allow movement of water and cations, but the nonpolar matrix not conducting anions) belong to this general context.

FIG. 1:

(a) Cartoon of two pathways for ion conduction in nanoporous materials. (b) Sketch of counterion conduction pathways in a semidilute polyelectrolyte solution of mesh size ξ. A chemical example of two adjacent repeat units with charge separation λ₀ is illustrated in the expanded scale. (c) Cartoon of percolating aggregates from ionic clusters constituting the skeletons, where the matrix is permeable to the conducting ion with an intrinsic ionic conductivity σ₀. The average distance between two adjacent ion clusters is λ and the mesh size for the percolating aggregate is ξ. (d) The free energy profile F_ion corresponding to sequential unbinding and binding of the conducting ion is periodic, with a free energy barrier U and period λ. The electric energy gain due to externally imposed electric field E is F_electric = −QEx, where Q is the charge of the ion and x is the location of the ion in the direction of the electric field.

The second scenario is ion conduction in polyelectrolyte solutions. As cartooned in Fig.1b, let us consider semidilute solutions of uniformly charged polyelectrolyte chains where the charge separation along the chain backbone is λ₀ and the correlation length for monomer concentration (mesh size) is ξ. As in the first scenario, ion conduction occurs via the two mechanisms of counterion-hopping along the chain backbone and permeation through the solvent. A typical trajectory of a counterion under an external electric field is a combination of hopping to neighboring sites with spacing λ₀ and permeation through an average mesh size ξ (Fig.1b). Such a situation is also relevant to solutions of polymerized ionic liquids.

The third scenario is in the context of single-ion conductive ionomers, where many clusters of ion-pairs (dipoles) exist as percolated aggregates in a conductive polymer matrix. Examples of the matrix are poly(ethylene oxide) and polycarboxylates and their chemical modifications, and gel polymer electrolytes, which are intrinsically permeable to the conducting ion with ion conductivity σ₀. The percolated aggregates are composed of clusters of ion-pairs formed by multiple chains. Unlike the situation of polyelectrolyte chains, where the ion binding is typically a single ion-pair formation, there are many ion-pairs (arising from both intra-chain and inter-chain interactions) in each of the ion clusters. Let the average distance between adjacent ionic clusters along a skeleton of the percolating aggregate be λ and the average mesh size in the background conductive polymer matrix be ξ (Fig.1c). The distance λ between ion clusters constituting a skeleton is related in a complex manner to the spacing of charged groups on the parent polymer chain (analogous to λ₀, but now the spacer moiety is also conductive).

In the context of the above examples, we address ion conduction due to a combination of ion-hopping along a chain or surface (Figs.1a,b) or skeleton of ion clusters (Fig.1c), and permeation through a conductive matrix. There are two length scales that characterize the morphology of the above conducting systems: the average hopping distance λ and the average permeation length ξ. In addition to the these variables characterizing the morphology of the system, the energy U to unbind X⁺ (Fig.1b) or its analog in an ion cluster from its corresponding anion is an important variable that controls ionic conductivity due to X⁺. In the simplest situation, U for monovalent charges is U = e²/(4πϵ₀ϵ_effr), where e is the electronic charge, ϵ₀ is the permittivity of vacuum, ϵ_eff is the effective dielectric constant in the local environment around the bound X⁺, and r is the inter-ion distance in an ion-pair. In the case of ionic clusters, U arises from collective interactions among all ion-pairs inside a cluster and we absorb this important effect through the local effective dielectric constant, which is different from that in the matrix.

Following the law for rates of activated processes, the time τ required to unbind the conducting ion from the ion-pair is τ = τ₀ exp(U/k_BT), where τ₀ is the characteristic time for a free conducting ion to diffuse a distance comparable to its linear size, and k_BT is the Boltzmann constant times the absolute temperature. For the purpose of quoting later, the above expressions for U and τ are repeated as

$$U=\frac{e^2}{4\pi {ϵ}_0{ϵ}_{\text{eff}}r},\hspace{1em}\tau ={\tau }_0\text{exp}\left(\frac{U}{k_{B}T}\right).$$ (1)

In general, the value of τ can be additionally influenced by local segmental dynamics of the polymer as well as collective phonon dynamics of the system.

Consider an 1-dimensional ionic conductor where the binding sites for the drifting ion are uniformly present with a period λ. When the applied electric field is not too strong, the moving ion undergoes a series of binding and unbinding at every binding site. This process continues periodically until the ion reaches its favored electrode. Let us denote the free energy profile associated with unbinding of an ion at one location and binding back at the next neighboring binding site at a distance λ as a symmetric triangular free energy barrier of height U (Fig.1d). Therefore, the free energy landscape F_ion(x) due to binding and unbinding along the direction x of the constant electric field E is as depicted in Fig.1d with a period λ. The electric contribution to the free energy profile of the ion is F_electric(x) = −QEx, where Q = z₊e (z₊ is the valency of the conducting ion). The net free energy profile is

$$F(x)=F_{\text{ion}}(x)+F_{\text{electric}}(x).$$ (2)

In addition to the 1-dimensional hopping pathway for ion conduction (Fig.1d), we allow a second pathway of ion permeation from one linear assembly of binding sites to another linear assembly of binding sites as cartooned in Fig.2a. In the case of polyelectrolyte solutions, these two pathways can also be labelled as ‘intra-chain’ and ‘inter-chain’ conduction pathways, respectively; for ionic aggregates, these are, respectively, intra-skeleton and inter-skeleton pathways. Assessment of the relative contributions from the intra- and inter-conduction pathways is of considerable interest in the general contexts mentioned in Fig.1.

FIG. 2:

(a) Sketch of the permeation mechanism over a distance ξ through the matrix that occurs in parallel to 1-dimensional hopping conduction over a distance mλ. (b) The corresponding free energy landscape, where L is the period, λ is the sub-period, U is the barrier height, and ξ is the hopping distance.

In view of the above considerations, we present an analytically tractable theory for the model sketched in Fig.2a, where the curved lines represent hopping pathways and empty space represent permeation pathways. Let there be an infinite number of periods of length L, where each period is labelled by the index N ≥ 1. In each period, there are m sub-periods for 1-dimensional conduction, followed by a 3-dimensional hopping over a distance ξ through the matrix. Let the regular sub-period for 1-dimensional unbinding-binding process be λ. Since ion hopping occurs along the curvilinear contour, we take λ as the average distance between adjacent binding sites projected on the direction of the external electric field. Let n denote the label of the sub-period, 1 ≤ n ≤ m. In each sub-period, we take the free energy profile as triangular with barrier height U, as shown in Fig.2a. The periodic free energy profile in the absence of the external electric field is given as

$$F_{\text{ion}}(NL+x)=F_{\text{ion}}(x),\hspace{1em}N\ge 1$$ (3)

and for each N ≥ 1,

$$F_{\text{ion}}(n\lambda +x)=F_{\text{ion}}(x).\hspace{1em}1\le n<m,0<x<\lambda$$ (4)

In addition to the periodic profile, the applied electric field gives the down hill free energy contribution −QEx. Adding F_ion(x) and F_electric(x), the free energy profile for the first period follows as

$$F(x)=\{\begin{array}{ll}-\frac{2U}{\lambda }[x-(n-1)\lambda ]-QEx,\hfill & (n-1)\lambda \le x\le \left(n-\frac{1}{2}\right)\lambda \hfill \\ -2U+\frac{2U}{\lambda }[x-(n-1)\lambda ]-QEx,\hfill & \left(n-\frac{1}{2}\right)\lambda \le x\le n\lambda \hfill \\ -QEx\hfill & m\lambda \le x\le L=m\lambda +\xi \hfill \end{array}$$ (5)

The same result is applicable to other periods as well, with x in the electric field contribution taken as the distance from the origin.

The Langevin equation for the dynamics of the ion at position x and time t is given as

$$\zeta \frac{dx}{dt}=-\frac{\partial F(x)}{\partial x}+\sqrt{\zeta k_{B}T}\Gamma (t),$$ (6)

where ζ is the friction coefficient of the ion in the matrix and Γ(t) is the random noise taken to be white noise satisfying the fluctuation-dissipation theorem,

$$\langle \Gamma (t)\rangle =0,\hspace{1em}\langle \Gamma (t)\Gamma \left(t^{\prime }\right)\rangle =2\delta \left(t-t^{\prime }\right),$$ (7)

where the angular brackets denote the averaging over the probability distribution function P(x, t) of finding the ion at position x and time t. Identifying the diffusion coefficient D of the ion in the matrix as D = k_BT/ζ, Eq.(6) becomes

$$\frac{dx}{dt}=-D\frac{\partial F(x)/k_{B}T}{\partial x}+\sqrt{D}\Gamma (t).$$ (8)

Performing the average over P(x, t), the above equation gives the average velocity v as

$$v=\langle \frac{dx}{dt}\rangle =-D\langle \frac{\partial }{\partial x}\left(F(x)/k_{B}T\right)\rangle ,$$ (9)

where 〈Γ(t)〉 = 0 from Eq.(7) is used. Note that, if there are no barriers and there is only electrophoretic drift, F(x) = −QEx, then $v=\frac{QD}{k_{B}T}E$ so that the ionic conductivity σ₀ = Qv/E is given by Q²D/k_BT. In the presence of barriers, the ionic conductivity σ is modified from this result and can generally be represented in terms of an effective diffusion coefficient D_eff(U, E) which depends on the barrier height U and E,

$${\sigma }_0=\frac{Q^{2}D}{k_{B}T},\hspace{1em}\sigma =\frac{{\sigma }_0}{D_0}{D}_{\text{eff}}(U,E).$$ (10)

Note that D corresponds to ion diffusion in the matrix and it depends on the various ion hopping processes involving the barriers in the matrix, that are responsible for ion conduction in the matrix.

The derivation of σ is as follows. We calculate the average velocity v of the ion from Eq.(9) using P(x, t) and get σ from the relation σ = Qv/E. The probability distribution function P(x, t) follows from the Fokker-Planck formalism of the Langevin equation (Eq.(8)) as [33,34]

$$\frac{\partial P(x,t)}{\partial t}=-\frac{\partial }{\partial x}J(x,t),$$ (11)

$$J(x,t)=-D\left[\left(\frac{\partial F(x)/k_{B}T}{\partial x}\right)P(x,t)+\frac{\partial P(x,t)}{\partial x}\right].$$ (12)

In the steady state where the flux J is constant, integration of Eq.(12) from x = 0 gives

$$P(x)=e^{-F(x)/k_{B}T}\left[P(x=0)-\frac{J}{D}{\int }_0^{x}d{x}^{\prime }{e}^{F\left(x^{\prime }\right)/k_{B}T}\right].$$ (13)

P(x = 0) is determined from P(NL + x) given by the above equation. Using Eq.(5), the integral on the right-hand side of Eq.(13) becomes

$${\int }_0^{NL+x}d{x}^{\prime }{e}^{F\left(x^{\prime }\right)/k_{B}T}=I_{+}\frac{\left(1-e^{-\mathit{\text{NQEL}}/k_{B}T}\right)}{\left(1-e^{-QEL/k_{B}T}\right)}+e^{-\mathit{\text{NQEL}}/k_{B}T}{\int }_0^{x}d{x}^{\prime }{e}^{F\left(x^{\prime }\right)/k_{B}T},$$ (14)

where

$$I_{+}={\int }_0^{L}dx{e}^{F(x)/k_{B}T}.$$ (15)

Combining Eqs.(13) and (14), we get

$$P(NL+x)=e^{\mathit{\text{NQEL}}/k_{B}T-F(x)/k_{B}T}\left[P(x=0)-\frac{J}{D}\frac{I_{+}}{\left(1-e^{-QEL/k_{B}T}\right)}\right]+e^{-F(x)/k_{B}T}\left[\frac{J}{D}\frac{I_{+}}{\left(1-e^{-QEL/k_{B}T}\right)}-\frac{J}{D}{\int }_0^{x}d{x}^{\prime }{e}^{F\left(x^{\prime }\right)/k_{B}T}\right]$$ (16)

Since P(NL + x) must be finite for N → ∞,

$$P(x=0)=\frac{J}{D}\frac{I_{+}}{\left(1-e^{-QEL/k_{B}T}\right)},$$ (17)

and hence we get from Eqs.(13) and (16)

$$P(NL+x)=P(x).$$ (18)

Because of periodicity, P(x) is normalized in every period so that

$${\int }_0^{L}dxP(x)=L.$$ (19)

Substituting Eq.(17) into Eq.(13) and performing the integration from x = 0 to x = L, we get

$$J=DL\frac{\left(1-e^{-QEL/k_{B}T}\right)}{\left[I_{+}{I}_{-}-\left(1-e^{-QEL/k_{B}T}\right)Y\right]}$$ (20)

where

$$I_{-}={\int }_0^{L}dx{e}^{-F(x)/k_{B}T},$$ (21)

and

$$Y={\int }_0^{L}dx{e}^{-F(x)/k_{B}T}{\int }_0^{x}d{x}^{\prime }{e}^{F\left(x^{\prime }\right)/k_{B}T}.$$ (22)

The velocity of the ion follows from Eqs.(9) and (12) as

$$v=\frac{1}{L}{\int }_0^{L}dx\left[J+D\frac{\partial P(x)}{\partial x}\right]=J,$$ (23)

where the periodicity property P(L) = P(0) is used. The general expression for the ionic conductivity, σ = Qv/E, is given by Eqs.(20) and (23) as

$$\sigma =\frac{QDL}{E}\frac{\left(1-e^{-QEL/k_{B}T}\right)}{\left[I_{+}{I}_{-}-\left(1-e^{-QEL/k_{B}T}\right)Y\right]}.$$ (24)

where I₊, I₋, and Y are given in Eqs.(15), (21), and (22), respectively.

Focusing on the linear response regime (Ohm’s regime), namely QEλ<k_BT, the ionic conductivity is given by

$$\sigma =\frac{Q^{2}D{L}^2}{k_{B}T}\frac{1}{I_{+}(E\to 0)I_{-}(E\to 0)},$$ (25)

where I₊(E → 0) and I₋(E → 0) follow from Eqs.(15) and (21) as

$$I_{+}(E\to 0)=-m\lambda \left(\frac{k_{B}T}{U}\right)\left(e^{-U/k_{B}T}-1\right)+\xi ,$$ (26)

$$I_{-}(E\to 0)=-m\lambda \left(\frac{k_{B}T}{U}\right)\left(1-e^{U/k_{B}T}\right)+\xi .$$ (27)

Substitution of Eqs.(26) and (27) into Eq.(25) yields a general expression for ionic conductivity in the linear regime as

$$\frac{\sigma }{{\sigma }_0}={\left(1+\frac{\xi }{m\lambda }\right)}^2\frac{{\left(U/k_{B}T\right)}^2}{\left[e^{U/k_{B}T}-1+\frac{U}{k_{B}T}\left(\frac{\xi }{m\lambda }\right)\right]\left[1-e^{-U/k_{B}T}+\frac{U}{k_{B}T}\left(\frac{\xi }{m\lambda }\right)\right]}.$$ (28)

Note that the right-hand side can be interpreted as D_eff in units of D as defined in Eq.(10). As evident from Eq.(28), the ratio σ/σ₀ of ionic conductivity at T to that at T → ∞ depends on two key parameters, namely U/k_BT and ξ/mλ. As already noted, U/k_BT is dictated by the specificity of the ion-pairs. The second factor ξ/mλ is the partition ratio of the permeation length ξ through the matrix to the hopping distance along 1-dimensional trajectory. This partition ratio is a measure of structural heterogeneity in the material. For large barriers U/k_BT ≫ 1, Eq.(28) reduces to the limiting laws

$$\frac{\sigma }{{\sigma }_0}=\{\begin{array}{ll}{\left(\frac{U}{k_{B}T}\right)}^2{e}^{-U/k_{B}T}\hfill & \frac{\xi }{m\lambda }=0,\frac{U}{k_{B}T}\gg 1\hfill \\ \left(\frac{\xi }{m\lambda }\right)\left(\frac{U}{k_{B}T}\right)e^{-U/k_{B}T}\hfill & \frac{\xi }{m\lambda }\gg 1,\frac{U}{k_{B}T}\gg 1\hfill \end{array}$$ (29)

Evidently, the temperature dependence of ionic conductivity is not simply the Arrhenius behavior due to the presence of the prefactors that depend on the barrier height.

The dependence of σ/σ₀ on U and the partition ratio ξ/mλ given by Eq.(28) is presented in Fig.3a. The extent of decrease in σ/σ₀ with an increase in U/k_BT depends on the partition ratio. As seen in Fig.3a, σ/σ₀ first decreases with ξ/mλ and then increases for higher values of ξ/mλ. This nonmonotonic behavior is illustrated in Fig.3b for U = 20k_BT. The initial decrease is due to the inefficiency of the permeation pathway for such short ξ, namely the loosened ion from its binding site immediately returns back to the same binding site. For larger permeation distances, the ion escapes from the binding traps along its original path. The minimum value of σ/σ₀ occurs at (ξ/mλ)^⋆ given by

$${\left(\frac{\xi }{m\lambda }\right)}^{\star }=\frac{(a+b)\left(U/k_{B}T\right)-2ab}{(a+b)\left(U/k_{B}T\right)-2{\left(U/k_{B}T\right)}^2},$$ (30)

where a and b are defined as

$$a=\left(1-e^{-U/k_{B}T}\right),\hspace{1em}b=\left(e^{U/k_{B}T}-1\right).$$ (31)

For large barriers as illustrated in Fig.3b, ${\left(\frac{\xi }{m\lambda }\right)}^{\star }\to 1$.

FIG. 3:

(a) Plot of σ/σ₀ given by Eq.(28) versus U/k_BT for different values of the partition ratio ξ/mλ. (b) Nonmonotonic dependence of σ/σ₀ on the partition ratio for U = 20k_BT. Beyond the critical value (ξ/mλ)_c, ionic conductivity is enhanced compared to the case of no permeation pathway.

As the partition ratio is increased from ${\left(\frac{\xi }{m\lambda }\right)}^{\star }$, there exists a threshold value ${\left(\frac{\xi }{m\lambda }\right)}_c$ beyond which the conductivity is above the value when the permeation mechanism is absent. This is shown in Fig.3b for U = 20k_BT. In general, the dependence of ${\left(\frac{\xi }{m\lambda }\right)}_c$ on U/k_BT is given by

$${\left(\frac{\xi }{m\lambda }\right)}_c=\frac{2ab-(a+b)U/k_{B}T}{{\left(U/k_{B}T\right)}^2-ab},$$ (32)

where a and b are given in Eq.(31). This result is plotted in Fig.4a, which delineates the regime of enhanced conductivity (ξ/mλ>(ξ/mλ)_c) and lowered conductivity (ξ/mλ<(ξ/mλ)_c) due to access to the permeation mechanism. For large barriers, (ξ/mλ)_c ≃ U/k_BT, consistent with the crossover behavior between the two limits given in Eq.(29).

FIG. 4:

(a) Plot of the threshold value (ξ/mλ)_c versus U/k_BT. For partition ratios larger than the threshold value ionic conductivity is enhanced, and for partition ratios below the threshold value, conductivity is lowered. (b) Enhancement of ionic conductivity by designing chemical routes to interrupt 1-dimensional conduction at mλ by creating a poison and to force the ion into permeation mechanism.

The prediction from Eq.(28) and Fig.4a provides a design strategy to significantly enhance ionic conductivity in heterogeneous structures. For example, if the 1-dimensional conductive pathway is blocked at mλ by doping the polymer chain with a non-conductive segment which functions as a poison (Fig.4b), then the permeation mechanism is forced on the conducting ion resulting in enhanced conduction for ξ/mλ>(ξ/mλ)_c.

In summary, a general theory for ionic conductivity in heterogeneous polymer structures is derived in terms of the energy U to unbind a conductive ion from the polymer backbone and the partition ratio ξ/mλ between 3-dimensional permeation and 1-dimensional conduction. For large barriers, the conductivity is pseudo-Arrhenius, with the prefactor of the exponential Arrhenius term depending on U and ξ/mλ, as given in Eq.(29). The prefactor can significantly affect the deduction of the activation barrier based on the standard practice of using the Arrhenius form. For example, according to the Arrhenius form, the temperature dependence of log₁₀(σ/σ₀) versus 1/T is given by the bottom line in Fig.5 for U = 100 kJ/mol and ξ/mλ = 0. On the other hand, the use of Eq.(28) gives the intermediate curve for the same values of U and ξ/mλ as for the bottom line. The conductivity is higher by more than an order of magnitude. If ξ/mλ>(ξ/mλ)_c, then the conductivity is even higher, as illustrated by the top curve for ξ/mλ = 100. If the middle curve were to be fitted with the Arrhenius form, the inferred U is approximately 75 kJ/mol. Therefore, the true free energy barrier is actually higher than the value obtained from direct implementation of the Arrhenius form.

FIG. 5:

Plot of σ/σ₀ versus 1/T for U=100 kJ/mol. Bottom line: Arrhenius plot; middle and top curves are from Eq.(28) with ξ/mλ = 0, and 100, respectively.

The limits given in Eq.(29) can be equivalently represented in terms of the unbinding time τ (Eq.(1)) as

$$\frac{\sigma }{{\sigma }_0}={\left(\frac{\tau }{{\tau }_0}\right)}^{-1}{\text{ln}}^y\left(\frac{\tau }{{\tau }_0}\right),$$ (33)

where y = 1 and 2 for ξ/mλ ≫ 1 and ξ/mλ = 0, respectively. The value of the exponent y can be slightly different if the free energy profile in every sub-period is smoother than the triangular profile used here. Ignoring the weak logarithmic corrections, we get

$$\sigma ~{\tau }^{-1}.$$ (34)

This general prediction valid for large barriers is consistent with prior simulation and experimental results in the literature [12, 26, 30]. If the barrier is weak, the full expression given in Eq.(28) needs to be used in conjunction with Eq.(1) to obtain the τ-dependence of the ionic conductivity.

Although the theory is presently couched in the context of single-ion conductors in dense heterogeneous polymer systems, it is applicable to polyelectrolyte solutions and polymerized ionic liquids as well where inter-chain hopping mechanism is prevalent in addition to intra-chain ion conduction. Additional features to the present model such as local segmental dynamics and collective phonon modes can be of importance. Assessment of contributions from these effects to ionic conductivity is relegated to future work.