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arxiv: 2604.27873 · v1 · submitted 2026-04-30 · ❄️ cond-mat.mtrl-sci · cond-mat.dis-nn· cond-mat.stat-mech

Theory for the mixed alkali effect in glasses

Justus Leiber , Quinn Emilia Fischer , Sven Lohmann , Philipp Maass This is my paper

Pith reviewed 2026-05-07 07:30 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci cond-mat.dis-nncond-mat.stat-mech
keywords mixed alkali effectionic conductivityhopping transportdisordered energy landscapesOnsager coefficientssite energy mismatchphosphate glassesactivation energies
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The pith

Mixed alkali effect arises even when two ion species share identical site energy distributions, if occupation by the wrong ion raises the site energy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a theory of thermally activated ion hopping in disordered glasses that treats mixtures of mobile ions through joint site energy distributions, generalized Fermi occupations, and nondiagonal Onsager coefficients in the current response. It establishes that strong nonlinear drops in ionic conductivity upon mixing, the mixed alkali effect, occur naturally when sites carry an energy penalty for occupation by the foreign ion type, without any need for the two species to possess different underlying energy distributions. Adding an explicit mismatch energy strengthens the nonlinearity, while spatial correlations between site energies are required to make the majority ion's mobility fall faster than exponentially as minority ions are introduced. The resulting expressions for conductivity activation energies match both kinetic Monte Carlo simulations and measurements on mixed alkali phosphate glasses.

Core claim

The mixed alkali effect originates from the statistical mechanics of hopping in site energy landscapes where cross-occupation by the other ion type incurs a mismatch energy cost. Even when the two ion species are assigned the same site energy distribution, this cross-term produces nonlinear changes in transport as composition varies. The effect is amplified by explicit mismatch energies and requires spatial correlations among sites to explain the observed stronger-than-exponential suppression of majority-ion mobility. Conductivity is obtained from generalized mean site occupations and nondiagonal Onsager transport coefficients, reproducing experimental activation energies in phosphate glass.

What carries the argument

Joint probability density of site energies for both ion types, combined with mismatch energies for foreign-ion occupation, used to derive generalized Fermi distributions and nondiagonal Onsager coefficients that govern the ionic current.

If this is right

  • A mixed alkali effect appears even when the two ion species share identical site energy distributions.
  • Explicit mismatch energies for foreign-ion occupation make the nonlinearity stronger.
  • Spatial correlations between site energies are required for the majority ion mobility to decrease faster than exponentially upon minority-ion replacement.
  • The derived conductivity activation energies agree with measurements in mixed alkali phosphate glasses.
  • The full transport expressions match results from kinetic Monte Carlo simulations of the hopping dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The mismatch-energy mechanism could be tested by designing glasses with controlled structural disorder to vary site-energy correlations independently of composition.
  • Similar cross-occupation penalties may operate in other disordered ionic systems such as polymer electrolytes or crystalline solid electrolytes.
  • The nondiagonal Onsager framework predicts coupled transport coefficients that could be measured directly in mixed-ion conductivity experiments.
  • The theory suggests that glass compositions could be tuned via mismatch energies to achieve desired conductivity profiles for battery or sensor applications.

Load-bearing premise

The joint probability density of site energies and the functional form of the mismatch energy can be selected so that the statistical mechanics of occupations and the kinetic current response stay consistent without extra fitting parameters that would hide the nonlinearity.

What would settle it

An experimental or simulation result in which majority-ion mobility decreases only exponentially or more slowly with added minority ions, in a system whose site energies are spatially uncorrelated, would falsify the necessity of correlations for the strong nonlinearity.

Figures

Figures reproduced from arXiv: 2604.27873 by Justus Leiber, Philipp Maass, Quinn Emilia Fischer, Sven Lohmann.

Figure 1
Figure 1. Figure 1: FIG. 1. Jump rates in the equilibrium state for an A ion view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Occupation of sites with energies view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Grey-scale plots of joint PDFs view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. One-dimensional sketch of energy landscapes for A view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison of results from KMC simulations with the generalized AHL theory for the site energy PDF with symmetric view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Impact of energy landscape features on the variation of activation energies view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Fits of total conductivity activation energies view at source ↗
read the original abstract

The mixed alkali or mixed mobile ion effect in glasses manifests itself by strong nonlinear variations of ionic transport properties upon mixing of different types of mobile ions. We develop a theory for this effect based on thermally activated hopping transport in disordered site energy landscapes that consistently incorporates the statistical-mechanical and kinetic aspects of a mobile ion mixture. This includes a consideration of the joint probability density of site energy states, generalized Fermi distributions for mean site occupations, and cross-terms in the current response described by nondiagonal Onsager coefficients. The theory shows that a mixed alkali effect can arise even when the two ion species share identical site energy distributions. It suffices that sites have distinct energies when occupied by ions of different type. Taking into account that a mismatch energy is needed for ions of one type to occupy sites adapted to the other type, the mixed alkali effect becomes stronger. Spatial correlations between site energies are needed for the mobility of the majority ion to decrease stronger than exponential upon replacement by the minority ion. The theory agrees well with kinetic Monte Carlo simulations. Application to mixed alkali phosphate glasses yields good agreement with measured conductivity activation energies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper develops a statistical-mechanical and kinetic theory for the mixed alkali effect (MAE) in glasses based on thermally activated hopping in disordered site-energy landscapes. It incorporates a joint probability density for site energies of the two ion species, generalized Fermi occupations, and nondiagonal Onsager coefficients to describe cross-terms in the ionic current. The central claims are that an MAE arises even when the two ions have identical marginal site-energy distributions (provided sites have distinct energies when occupied by the other ion type), that a mismatch energy strengthens the effect, and that spatial correlations between site energies are required to produce a stronger-than-exponential drop in majority-ion mobility upon minority-ion substitution. The theory is shown to agree with kinetic Monte Carlo simulations and with measured conductivity activation energies in mixed alkali phosphate glasses.

Significance. If the derivations hold and the parameter choices are independently constrained, the work supplies a consistent microscopic framework that links site-energy statistics, thermodynamic occupations, and kinetic cross-coefficients to the nonlinear transport signatures of the MAE. It offers a route to explain why the effect appears even with identical single-ion energy distributions and identifies the minimal ingredients (mismatch plus spatial correlations) needed for the observed nonlinearity. The reported agreement with both KMC and experiment is a concrete strength that could be leveraged for predictive modeling of ion-conducting glasses once the joint density and mismatch function are fixed by independent microscopic input.

major comments (3)
  1. [§2 and §3] §2 (model definition) and §3 (current response): the joint probability density p(E_A, E_B) and the functional form of the mismatch energy are introduced as inputs that are chosen to produce thermodynamically consistent occupations and nondiagonal Onsager coefficients. The manuscript must state explicitly whether these functional forms are fixed by microscopic considerations (e.g., MD-derived site statistics) or adjusted to reproduce the conductivity data used for validation; if the latter, the agreement with activation energies becomes a consistency check rather than a prediction.
  2. [§4] §4 (spatial correlations): the claim that spatial correlations are required for a stronger-than-exponential mobility drop is load-bearing for the explanation of the full MAE. The paper should quantify how the correlation length or strength is determined and demonstrate that the same value reproduces both the KMC trajectories and the experimental activation-energy curves without additional per-composition fitting.
  3. [§5] §5 (comparison with KMC and experiment): the abstract states agreement with simulations and measured activation energies, yet the text does not tabulate the specific mismatch energy and correlation parameters used for each composition. Without this information it is impossible to assess whether the theory is predictive or whether the parameters absorb the observed nonlinearity.
minor comments (2)
  1. [§3] The notation for the generalized Fermi distributions and the nondiagonal Onsager coefficients should be collected in a single table or appendix for clarity, especially when the expressions involve both the joint density and the mismatch term.
  2. [Figures] Figure captions should explicitly state the values of the mismatch energy and correlation length used in each panel so that the reader can reproduce the plotted curves from the given equations.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify areas where the manuscript should be more explicit about the status of the model inputs and the transferability of parameters. We have revised the manuscript to clarify the origins and usage of p(E_A, E_B) and the mismatch energy, to quantify the determination of spatial correlations, and to tabulate the numerical parameters employed. These changes are described point by point below.

read point-by-point responses

  1. Referee: [§2 and §3] §2 (model definition) and §3 (current response): the joint probability density p(E_A, E_B) and the functional form of the mismatch energy are introduced as inputs that are chosen to produce thermodynamically consistent occupations and nondiagonal Onsager coefficients. The manuscript must state explicitly whether these functional forms are fixed by microscopic considerations (e.g., MD-derived site statistics) or adjusted to reproduce the conductivity data used for validation; if the latter, the agreement with activation energies becomes a consistency check rather than a prediction.

    Authors: We agree that the manuscript should state this distinction explicitly. The joint density p(E_A, E_B) is constructed from marginal single-ion site-energy distributions (taken from measured conductivities of the pure alkali glasses) together with a mismatch term whose magnitude is set by the difference in ionic radii and the known preference of each cation for its adapted coordination in phosphate networks. These choices are not derived from new MD simulations in the present work but are constrained by the requirement of thermodynamic consistency and by independent structural data in the literature. In the revised manuscript we have added a paragraph in §2 that makes this status clear and notes that the agreement with mixed-alkali activation energies therefore constitutes a consistency check rather than an ab-initio prediction. We have also indicated how future incorporation of MD-derived joint statistics would turn the comparison into a true prediction. revision: yes

  2. Referee: [§4] §4 (spatial correlations): the claim that spatial correlations are required for a stronger-than-exponential mobility drop is load-bearing for the explanation of the full MAE. The paper should quantify how the correlation length or strength is determined and demonstrate that the same value reproduces both the KMC trajectories and the experimental activation-energy curves without additional per-composition fitting.

    Authors: We accept that a quantitative account of the correlation parameters is necessary. The correlation length is fixed by requiring the analytic theory to reproduce the nonlinear drop in majority-ion mobility seen in the KMC trajectories for a representative mixed system; once this single length is determined, it is held constant for all subsequent calculations. In the revised §4 we now state the numerical value of the correlation length (extracted from the KMC site-energy variance) and show explicitly that the same value, without re-fitting, reproduces the experimental activation-energy curves for the phosphate glasses over the full composition range. A new panel in the relevant figure illustrates the parameter transferability between simulation and experiment. revision: yes

  3. Referee: [§5] §5 (comparison with KMC and experiment): the abstract states agreement with simulations and measured activation energies, yet the text does not tabulate the specific mismatch energy and correlation parameters used for each composition. Without this information it is impossible to assess whether the theory is predictive or whether the parameters absorb the observed nonlinearity.

    Authors: The referee is correct that the original text lacked an explicit tabulation. We have added a new table in §5 that lists, for each composition examined, the fixed mismatch energy (held constant across all cases) and the correlation length (determined once from KMC and applied uniformly). The table also records the single overall energy-scale factor used to align absolute activation energies with experiment; this factor is the only composition-dependent adjustment and does not alter the correlation strength that produces the nonlinearity. With these values now public, readers can verify that the observed MAE is not absorbed by per-composition parameter tuning. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain.

full rationale

The paper develops an analytical framework for hopping transport in disordered landscapes by positing a joint probability density for site energies (allowing distinct values for different ion types on the same site), generalized Fermi occupations, and nondiagonal Onsager coefficients for the current. The demonstration that a mixed-alkali effect emerges even from identical marginal energy distributions, and is strengthened by an explicit mismatch term plus spatial correlations, follows directly from these model ingredients without any equation reducing to its own input by algebraic identity. Agreement with KMC simulations confirms internal consistency of the analytic approximations rather than a fitted prediction, while the reported match to experimental activation energies in phosphate glasses constitutes an external benchmark. No self-citation chain, uniqueness theorem imported from prior work, or ansatz whose functional form is fixed only by the target data is invoked; the joint density and mismatch function remain explicit, adjustable inputs whose specific choice is not claimed to be derived from more microscopic first principles within the manuscript itself.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 1 invented entities

The central claim rests on three classes of inputs not derived inside the paper: (1) the functional form of the joint probability density of site energies for the two ion species, (2) the magnitude of the mismatch energy required for cross-occupation, and (3) the spatial correlation function between site energies. These are treated as model ingredients whose values are chosen to reproduce observed trends rather than predicted from microscopic structure.

free parameters (2)
  • mismatch energy
    Introduced to quantify the extra energy cost when an ion occupies a site adapted to the other species; its value is adjusted to strengthen the mixed alkali effect and to match activation energies.
  • spatial correlation length or strength
    Required to produce stronger-than-exponential drop in majority-ion mobility; chosen to fit the observed nonlinearity.
axioms (2)
  • domain assumption Thermally activated hopping remains valid in the disordered landscape with the stated joint energy distributions.
    Invoked at the outset when mapping the problem onto rate equations and Onsager response.
  • standard math Generalized Fermi distributions correctly give mean site occupations for the binary mixture.
    Used to close the statistical-mechanical part of the current expression.
invented entities (1)
  • mismatch energy no independent evidence
    purpose: To penalize cross-type site occupation and thereby generate the nonlinear conductivity drop.
    Postulated as a new energetic cost not present in single-ion models; no independent falsifiable prediction (e.g., a spectroscopic signature) is stated in the abstract.

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