1

Parameter Sensitivity Analysis to Improve Material Design for Novel

Zn-MnO2 Batteries with Ionic Liquid ElectrolytesZachary T. Gima & Bernard J. Kim

CE 295: Spring 2016

Abstract—Battery design at the materials level is often aninefficient process that requires many iterations and significantdevelopment costs. To improve the design process, parametersensitivity analysis can be performed to provide battery materialsscientists with better design intiution. A parameter sensitivityanalysis was developed for electrode parameters by adaptingthe single particle model (SPM) for a Zn-MnO2 chemistry.Parameters of interest included particle radius and electrodethickness. The parameter sensitivity analysis determined therelative sensitivity of the battery’s output voltage to theseaforementioned parameters as well as their linear dependenceon one another.

I. INTRODUCTION

A. Motivation & Background

The Advanced Manufacturing for Energy lab at UC Berke-ley is developing a printable, secondary Zn-MnO2 battery forInternet of Things (IoT) and grid-scale energy storage applica-tions (Figure 1). The battery utilizes an ionic liquid electrolytethat enables facile recharging of zinc and could potentiallylead to displacement of lithium-based systems in applicationswhere safety and cost considerations are paramount. Specif-ically, applications involving sensors for food or wearableswould significantly benefit from a safer and less hazardousenergy storage system.

Fig. 1. Cross section SEM of Zn-MnO2 battery developed by Advanced Man-ufacturing for Energy lab. The distinction between electrode and electrolytelayers is clearly visible, along with some individual particles.

Zachary T. Gima and Bernard J. Kim are with Mechanical Engineering,University of Californa, Berkeley, CA 94720, USA.Email: {ztakeo, bernard.kim}@berkeley.edu

This class of zinc-based secondary batteries with ionicliquid electrolytes is not yet well understood or characterized.Because this research is relatively recent, there exists muchroom for improvement beyond proof-of-concept performance[1]. In order to streamline experimental design, we desire amodel that can parameterize controllable battery characteris-tics (e.g. geometry, composition, etc.) and predict resultingsystem performance. Specifically, this model will inform keymanufacturing decisions in order to best improve the bat-tery’s output voltage. Devices for IoT demand certain voltageminimums in order to operate electronic components, andimproving the battery’s system voltage will help the systemmeet this minimum over a larger range of charge.

This study utilizes the single particle model (SPM) [2],[3] for its simplicity over more sophisticated models (such asDoyle-Fuller-Newman [4]). The model is accurate for low C-rates, which are typical of our battery. The SPM models eachelectrode as a single porous spherical particle and assumesconstant ion concentration within the electrolyte with respectto space and time [2], [3]. When applied to our battery, thismodel will help optimize electrode composition in order tofurther improve overall performance.

B. Relevant Literature

Lithium-ion batteries have emerged as the battery of choicefor portable electronics, as well as in vehicular and aerospaceapplications [5], [6]. In order to fully utilize the capabilityof these batteries, a battery management system must bedeployed with the battery to ensure safe and optimal operationfor a given application [5]. More sophisticated versions ofthese management systems employ an electrochemical modeldescribing the battery’s dynamics to monitor key performanceparameters such as state-of-charge and internal resistance [7].However, if the system’s dynamics are not well understood,then the model may provide insight into the battery’s behaviorunder certain load conditions.

The single particle model (SPM) was derived from a sim-plified version of the complete Doyle-Fuller-Newman model[3], [4] and has been widely utilized to model and understandlithium-ion batteries [2], [3], [7]. While the assumptions theSPM makes limit the model’s applicability to scenarios withlow C-rates or negligible electrolyte dynamics, its simplicityand ease of solving compared to more rigorous models [4]makes it an attractive tool, especially for more fundamentalanalyses of electrochemical dynamics.

The Zn-MnO2 battery of interest shares properties withpreviously modeled intercalation lithium-ion batteries that

2

make the SPM suitable to modeling this system. It has beendetermined that the main mechanism for charge storage inthis system is with intercalation of Zn2+ ions into the MnO2

cathode [8]. Furthermore, the electrolyte-agnostic assumptionsof the SPM also prove to be beneficial as this battery utilizesan ionic liquid electrolyte whose solvent behavior differssignificantly from traditional aqueous electrolytes [9]. The Zn-MnO2 battery is also being cycled at low C-rates, furtheragreeing with this assumption [10]. By adapting the SPM tothis Zn-MnO2 system, we hope to leverage parameter sensi-tivity analysis on the resulting model parameters to optimizecontrollable geometric and compositional factors to betterunderstand and improve the system.

C. Focus of this Study

This study will utilize the SPM to model the zinc-basedbattery in order to perform parameter sensitivity analysis ofthe battery’s output voltage to particle radius and electrodethickness. The results will be used to directly modify electrodegeometries and compositions in physical experiments to bettertailor performance outputs for IoT demands.

II. TECHNICAL DESCRIPTION

This study utilizes a reduced electrochemical battery model,the SPM. This model approximates each electrode of thebattery as collections of uniform particles. The dynamics ofone particle at each electrode are studied and then scaled up toreflect the volume of the entire electrode. This approximationresults from assuming that the concentration of insertion ionsin the electrolyte phase remains constant in space and timeand also that the entirety of the electrode evenly contributesto the total cell dynamics. The first assumption is reasonablyvalid for batteries charged and discharged at low C-rates. SeeFigure 2 for a graphical representation of the SPM.

For clarity, Table I provides definitions for all parametersrelated to the SPM that follow.

A. Single Particle Model

Specifically, the SPM results from applying the abovesimplifying assumptions to the more complex DFN model.The partial differential equations which therefore define theSPM come directly out of Fick’s Second Law of Diffusionin spherical coordinates for symmetric, 1-D diffusion at eachelectrode (1)-(4). These PDEs describe the dynamics of theinsertion ions in the solid phase.

@c

�s

@t

(r, t) = D

�s

2

r

@c

�s

@r

(r, t) +@

2c

�s

@r

2

�(1)

@c

+s

@t

(r, t) = D

+s

2

r

@c

+s

@r

(r, t) +@

2c

+s

@r

2

�(2)

@c

�s

@r

(0, t) = 0,@c

�s

@r

(R�s

, t) = � I(t)

D

�s

Fa

�AL

� (3)

@c

+s

@r

(0, t) = 0,@c

+s

@r

(R+s

, t) =I(t)

D

+s

Fa

+AL

+(4)

Fig. 2. Graphic representation of the SPM [3]. The electrodes are eachrepresented as single porous spherical particles. This results from assumingelectrolyte concentration remains constant in space and time.

TABLE ISINGLE PARTICLE MODEL PARAMETER DEFINITIONS

Symbol Description SI UnitsA Cell cross sectional area m2

a

j Specific interfacial surface area m2/m3

c

j

s

Concentration in solid phase mol/m3

c

j

ss

Concentration at particle surface mol/m3

c

j

s,max

Max concentration in solid phase mol/m3

D

j

s

Diffusion coefficient in solid phase m2/m3

F Faraday’s constant C/molI Input current Ai

j

0 Exchange current density Vj Positive (+) or negative (-) electrode -L

j Electrode thickness mR Universal gas constant J/mol-KR

f

Lumped current collector resistance ⌦R

j

s

Particle radius mr Radial coordinate m or m/mT Cell temperature Tt Time sec or sec/secU

j Equilibrium potential VV Output voltage V↵j Anodic/cathodic transfer coefficient -

The Neumann boundary conditions (3) and (4) at thesurfaces of the electrodes (r = R

±s

) proportionally relate theflux entering or exiting the electrode to the input current, I(t).The boundary conditions at the center of the electrodes (r =0) are required for well-posedness and spherical symmetry.

The system output is the cell voltage and is governed bya combination of Butler-Volmer kinetics, electrode thermody-namics, electrode OCP, and internal resistance (5).

3

V (t) =RT

↵F

sinh�1

I(t)

2a+AL

+i

+0

�c

+ss

(t)�!

� RT

↵F

sinh�1

I(t)

2a�AL

�i

�0

�c

�ss

(t)�!

+ U

+�c

+ss

(t)�� U

� �c

�ss

(t)�+R

f

I(t) (5)

The exchange current density i

j

0 and solid electrolyte surfaceconcentration c

j

ss

are described by (6) and (7). It should benoted that c

j

ss

simply describes the solid concentration of eachelectrode at the boundary of the spatial dimension, i.e. c

j

ss

(t)= c

j

s

(Rj

s

,t).

i

j

0

�c

j

ss

�= k

j

rc

0e

c

j

ss

(t)⇣c

j

s,max

� c

j

ss

(t)⌘

(6)

c

j

ss

(t) = c

j

s

�R

j

s

, t

�, j 2 {+,�} (7)

The full system plant is presented graphically in Figure 3.

Dynamic System | Σ

Parameters | "

#

Input Current

$

Output Voltage

State Variable | %&&

Fig. 3. Block diagram of system plant with input I(t) and output V(t).

B. SPM Normalization

The SPM equations (1)-(4) can be normalized in both timeand space in order to facilitate the sensitivity analysis, as willbe clarified later. This normalization is carried out by scalingthe radial r and time t coordinates by the particle radius andcharacteristic diffusion time respectively (8).

r =r

R

�s

, t =D

�s�

R

�s

�2 t (8)

Applying this normalization to the system equations (1)-(4)yields the following modified PDEs and boundary conditions(9)-(12).

@c

�s

@ t

(r, t) =2

r

@c

�s

@r

(r, t) +@

2c

�s

@r

2(9)

@c

+s

@ t

(r, t) =2

r

@c

+s

@r

(r, t) +@

2c

+s

@r

2(10)

@c

�s

@r

(0, t) = 0,@c

�s

@r

(1, t) = � R

�s

I(t)

D

�s

Fa

�AL

� (11)

@c

+s

@r

(0, t) = 0,@c

+s

@r

(1, t) =R

�s

I(t)

D

+s

Fa

+AL

+(12)

The notable change is that the particle radius, R

±s

now ap-pears explicitly in the boundary condition after normalization.

Henceforth, the bars above r and t will be dropped for clarityand brevity.

C. Sensitivity Equations

Next, we derive the sensitivity equations. We want to studythe influence of the parameters with respect to the systemoutput, not the states. That is, for a given change in ourparameters, we want to study how the system’s output voltagereacts (13).

@y

@✓

=@V

@✓

(13)

However, in order to calculate the sensitivity of the outputwith respect to the parameters, we must first find the sensitivityof the states with respect to the parameters. We can thencombine this with the derivative of the output with respectto the states by using chain rule (14).

@y

@✓

=@x

@✓

@y

@x

! @V

@✓

=@c

ss

@✓

@V

@c

ss

(14)

The parameters of interest to be studied are the particleradius R

±s

and the electrode thickness L

±. This results in fourseparate parameters for both the anode and cathode (15).

✓ =

2

664

✓

�1

✓

�2

✓

+1

✓

+2

3

775 =

2

664

R

�s

L

�

R

+s

L

+

3

775 (15)

The first step is to take the derivative of the output equationwith respect to the states. This equation is omitted for brevity.As can be noted from (5), the only state that the output voltageis dependent on is c

j

ss

, which is simply a boundary value ofc

j

s

.The second step is to derive the individual sensitivity

equations for each parameter. This is done by first evaluatingthe nominal state equation to determine the nominal solutionx(t,�0), evaluating the Jacobian matrices given by (16), andlastly solving the sensitivity equation (17) for S(t) [11].

A(t, �0) =@f(t, x, �)

@x

���x=x(t,�0),�=�0

,

B(t, �0) =@f(t, x, �)

@�

���x=x(t,�0),�=�0

(16)

S(t) = A(t, �0)S(t) +B(t, �0), S(t0) = 0 (17)

Therefore, the next step is to derive the sensitivity equationswith respect to the states. This is done by taking the derivativeof (9)-(12) with respect to ✓

i

for the appropriate electrodes(18).

S

i

(t, x, �) =@

@✓

i

@f(t, x, �)

@x

�(18)

The sensitivity equations for each parameter with respectto the states can then be computed. The sensitivity equationsw.r.t. ✓�1 = R

�s

are computed as in (19), (20), and (21).

4

S

�1t(r, t) = S

�1rr(r, t; ✓0) (19)

S

�1r(0, t) = 0 (20)

S

�1r(1, t) =

1

D

�s

Fa

�AL

� I(t) (21)

The sensitivity equations w.r.t. ✓�2 = L

� are computed as in(22), (23), and (24).

S

�2t(r, t) = S

�2rr(r, t; ✓0) (22)

S

�2r(0, t) = 0 (23)

S

�2r(1, t) = � R

�s

D

�s

Fa

�AL

�2I(t) (24)

The sensitivity equations w.r.t. ✓+1 = R

+s

are computed as in(25), (26), and (27).

S

+1t(r, t) = S

+1rr(r, t; ✓0) (25)

S

+1r(0, t) = 0 (26)

S

+1r(1, t) = � 1

D

+s

Fa

+AL

+I(t) (27)

The sensitivity equations w.r.t. ✓+2 = L

+ are computed as in(28), (29), and (30).

S

+2t(r, t) = S

+2rr(r, t; ✓0) (28)

S

+2r(0, t) = 0 (29)

S

+2r(1, t) =

R

+s

D

+s

Fa

+AL

�2I(t) (30)

With the sensitivity equations derived, we can now solvethese four sets of PDEs to compute S

�1 (r,t), S

�2 (r,t), S

+1 (r,t),

and S

+2 (r,t).

We are now positioned to apply the chain rule to getthe sensitivity of the output with respect to the states. Thatis, for every parameter, we can now multiply its sensitivityequation by @V

@cssto obtain our desired result (31). Note that

the sensitivity equations with respect to the state provide thesensitivity evolution with time for all spatial coordinates. Theappropriate value of S

i

(r,t) to use occurs at the boundary, i.e.

S

i

(1,t).

@V

@✓

j

i

=@V

@c

j

ss

· Si

(1, t) (31)

The final step is to normalize (31) with respect to eachparameter. Without this step, the values provided by thesensitivity vector become skewed by the magnitude of eachparameter, making it difficult to perform an objective analysisof each parameter’s influence on the output. The normalizationis performed by multiplying the final sensitivity vector by theparameter divided by the output (32).

@f

@✓

· ✓f

! @V

@✓

j

i

· ✓j

i

V

= S

⇤i

(32)

The results of (32) can then be gathered in a sensitivitymatrix S (33).

S =⇥S

�⇤1 S

�⇤2 S

+⇤1 S

+⇤2

⇤(33)

The sensitivity derivation is summarized in Figure 4.

Diffusion PDE

cs±(r,t)Output

V(css±)

css±(t)

· Si± (1,t)∂V∂css±

∂V∂θi±

I(t)

∂V∂css±

θi±V

·

Sensitivity PDE

Si±(r,t) Si±(1,t)

Fig. 4. Block diagram of sensitivity equation derivation. Only the boundaryvalues of c

±s

(r,t) and S

±i

(r,t) (cs

s±(r,t)) and S

±i

(1,t) are used in the finalsensitivity matrix.

D. Sensitivity Analysis

Let S

T

S = D

T

CD. The sensitivity analysis can then beperformed by decomposing the matrix S

T

S into its constituentsC and D. These two matrices are defined as follows (34).

D =

2

664

��S

�⇤1

�� 0 0 00

��S

�⇤2

�� 00 0

��S

+⇤1

�� 0 00 0 0

��S

+⇤2

��

3

775 ,

C =2

66666664

1 hS�⇤1 ,S

�⇤2 i

kS�⇤1 kkS�⇤

2 khS�⇤

1 ,S

�+⇤1 i

kS�⇤1 kkS+⇤

1 khS�⇤

1 ,S

+⇤2 i

kS�⇤1 kkS+⇤

2 khS�⇤

2 ,S

�⇤1 i

kS�⇤2 kkS�⇤

1 k 1 hS�⇤2 ,S

+⇤1 i

kS�⇤2 kkS+⇤

1 khS�⇤

2 ,S

+⇤2 i

kS�⇤2 kkS+⇤

2 khS+⇤

1 ,S

�⇤1 i

kS+⇤1 kkS�⇤

1 khS+⇤

1 ,S

�⇤2 i

kS+⇤1 kkS�⇤

2 k 1 hS+⇤1 ,S

+⇤2 i

kS+⇤1 kkS+⇤

2 khS+⇤

2 ,S

�⇤1 i

kS+⇤2 kkS�⇤

1 khS+⇤

2 ,S

�⇤2 i

kS+⇤2 kkS�⇤

2 khS+⇤

2 ,S

+⇤1 i

kS+⇤2 kkS+⇤

1 k 1

3

77777775

(34)Matrix D provides a direct quantification of parameter sen-

sitivity with larger numbers indicating higher sensitivity to thatparameter. Matrix C provides a measure of linear dependencebetween parameters. Values close to 1 and -1 indicate thatthose parameters are very strongly linearly dependent. Valuesequal to 1 and -1 indicate that the parameters are completelyproportional or inversely proportional.

III. DISCUSSION

A. Simulation

The PDEs were discretized for simulation using a combi-nation of second order central finite difference methods andfirst order forwards and backwards finite difference methodsfor the boundary conditions. The input current for the systemwas a pulsed discharge at 0.5C with a 50% duty cycle. Allparameter values used were for a LiCoO2 battery due to alack of comprehensive data for our Zn-MnO2 system.

Despite the mismatch of parameter values for this simula-tion, the results are still applicable for any battery system asthis sensitivity analysis relates the relative effects of differentparameters to each other. While the absolute magnitudes

5

present in the D matrix may not be representative, their relativevalues are relevant in describing the relationship betweenchosen parameters.

B. Results

The D and C matrices for the simulation ran are providedbelow (35).

D =

2

664

14.5850 0 0 00 6.8258 0 00 0 14.5850 00 0 0 6.8258

3

775 ,

C =

2

664

1 �1 �1 1�1 1 1 �1�1 1 1 �11 �1 �1 1

3

775

(35)

Again, for clarification, the parameter vector as in (15) isreproduced below.

✓ =

2

664

✓

�1

✓

�2

✓

+1

✓

+2

3

775 =

2

664

R

�s

L

�

R

+s

L

+

3

775

The values in D indicate that the particle radius is moreinfluential on the output voltage than the electrode thicknessfor both electrodes. For this particular simulation, this issubject to the parameters used and may change based ondifferent values for D

j

s

, ↵j , etc.The more significant result comes from the values present

in C. The entries corresponding to the linear dependence ofparticle radius and electrode thickness for each electrode areequal to -1. These values are expected because both parametersonly appear in the boundary condition within the sensitiv-ity equations (21), (27), (24), (30). This sensitivity analysissuggests that the particle radius and electrode thickness areinversely proportional to one another.

In reality, this inverse proportionality is not true because theSPM itself assumes that the dynamics occurring at a singleparticle can be scaled up to represent the entire volume ofeach electrode. For real batteries, the fraction of each electrodeparticipating in the reaction is limited by electrolyte dynamics,which are assumed away in the SPM, so changes in particleradius cannot be compensated for by changes in electrodethickness and vice versa.

However, this inverse proportionality does reveal that theparticle radius and electrode thickness are strongly linearlydependent, even if they are not completely proportional. Asa practical point, both parameters should not be changedsimultaneously to avoid obfuscating the effects of either onthe output voltage.

Ultimately, this analysis yields insight into which parametershould be prioritized for process control during manufacturing.Based on these results for the parameter values used, controlof particle size should be prioritized over electrode thicknesswhen seeking to tightly control output voltage.

C. Future Work

Future work should seek to add more insightful parametersthat still remain controllable during the manufacturing pro-cess. These can include the moles of cycleable zinc n

Zn,s

,the volume fraction of active material "

j

s

, and the specificinterfacial surface area ↵

j . Furthermore, the sensitivities ofadditional metrics (such as state of charge and cycle life)to these parameters should also be investigated. The SPMitself may need to be augmented by adding back in electrolytedynamics to more accurately model the true system dynamics[12].

IV. SUMMARY

A parameter sensitivity analysis was performed for bat-tery output voltage with respect to electrode parameters byadapting the single particle model (SPM). The parametersof interest were the particle radius and electrode thickness.Analysis concluded that output voltage is more sensitive toparticle radius than electrode thickness and that they areinversely proportional to each other, which is influenced by thesimplifying assumptions inherent in the SPM. The results rankthe importance of each parameter for use in manufacturingprocess control.

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