Modeling Volume Changes in Porous Electrodes

Journal of The Electrochemical Society, 53 A79-A86 2006 003-465/2005/53/A79/8/$20.00 The Electrochemical Society, Inc. Modeling olume Changes in Porous Electrodes Parthasarathy M. Gomadam*,a,z John W. Weidner** Center for Electrochemical Engineering, Deartment of Chemical Engineering, University of South Carolina, Columbia, South Carolina 29208, USA A79 A three-dimensional mathematical model is resented to describe volume changes in orous electrodes occurring during oeration. Material conservation equations are used to derive governing relationshis between electrode dimensions orosity for deosition/reciitation, intercalation, ionomer-based electrodes. By introducing a arameter, called the swelling coefficient, the relative magnitudes of the change in electrode dimensions the change in orosity are determined. The swelling coefficient is design-deendent measured exerimentally for a given cell design. The model is general forms a critical addition required to extend the existing orous electrode models to include volume change effects. For the secial case of uniform reaction distribution, analytical solutions are resented used to illustrate the effect of volume changes in orous electrodes. 2005 The Electrochemical Society. DOI: 0.49/.236087 All rights reserved. Manuscrit submitted July 29, 2005; revised manuscrit received Setember 22, 2005. Available electronically December 2, 2005. Many researchers have reorted significant volume changes occurring in orous electrodes. For examle, in Li SOCl 2 batteries reciitation of LiCl causes significant cathode swelling is considered an imortant henomenon governing the evolution of cell resistance during discharge. In Li-ion batteries, volume changes as high as 358% have been observed by Yang et al. 2 with Li Sn alloy intercalation anodes. Similarly, in roton exchange membrane PEM fuel cells the Nafion membrane as well as the ionomer hase in the catalyst layers swell with increase in water content, affecting cell resistance. 3 Mathematical models from the literature have considered volume changes in orous electrodes.,4-7 Many of these treatments include detailed electrochemistry nonuniform structural changes e.g., orosity changes. For examle, Alkire et al. 4 develoed a model for describing the nonuniform orosity changes occurring during anodic dissolution of a coer electrode. Assuming cylindrical ores, they analyzed the effects of orosity changes on the electrode resistance reaction current distribution. Dunning et al. 5 develoed a battery model to describe mass transfer of saringly soluble active material, which was extended by them 6 to describe the oeration of silver silver chloride cadmium cadmium hydroxide electrodes. Using this model they treated the effects of changing orosity reaction surface area in addition to charge mass transort occurring in the electrodes. A more detailed review of these other models considering structural changes in orous electrodes is given in Newman Thomas-Alyea. 7 However, as ointed out by these latter authors, 7 all the models in the literature considering structural changes in orous electrodes assume a constant geometric volume for the electrode throughout its oeration. Porosity is allowed to change during oeration but exansion or contraction is either ignored or built into the initial conditions. For examle, Jain et al. treated cathode swelling in their Li SOCl 2 battery model by using effective values for electrode thickness initial orosity. This method assumes constant electrode thickness during discharge resulting in significant errors in the electrode resistance. Further, their method assumes uniform swelling at all oints in the cathode, making it alicable only under conditions of nearuniform reaction distribution e.g., low-rate discharges. In Li-ion batteries, Botte 8 Christensen Newman 9 modeled volume changes occurring in a single host article but did not consider its effects on the volume changes of the orous electrode. In this aer we resent a three-dimensional mathematical model to describe volume changes occurring in orous electrodes. Based on material conservation equations we derive the governing equations relating the two effects of volume changes, namely, the change * Electrochemical Society Student Member. ** Electrochemical Society Active Member. a Present address: Medtronic Energy Comonent Center, Brooklyn Center, Minnesota 55430, USA. z E-mail: artha.m.gomadam@medtronic.com in the dimensions of the electrode the change in its orosity. A arameter, called the swelling coefficient, is introduced so as to determine the relative magnitudes of the two effects. For a given electrochemical cell the swelling coefficient is obtained exerimentally as discussed later. We consider three tyes of orous electrodes: a deosition/reciitation electrodes e.g., the carbon cathode in Li/SOCl 2 batteries, which exs as LiCl is deosited in the ores, b intercalation electrodes e.g., a Cu Sn alloy anode in Li-ion batteries, which exs as Li intercalates, c ionomerbased electrodes e.g., the catalyst layer in PEM fuel cells, which exs as water content of its ionomer hase increases. The model is general forms a critical addition required to extend the existing orous electrode models to include volume change effects. For the secial case of uniform reaction distribution, analytical solutions are resented used to illustrate the effect of volume changes in orous electrodes. Model Develoment Deosition/reciitation electrodes. Figure shows a schematic of a deosition/reciitation orous electrode. During oeration, a solid-hase reaction roduct forms on the surface of the electrode articles, occuying a art of the ore volume ushing the electrode articles away from each other. Thus, the volume change effects during oeration manifest in two forms: i change in electrode dimensions ii change in orosity. At all times, electronic connection is assumed to exist between the articles either through direct hysical contact between them or through the electronically conductive binder interconnections. The relationshi governing volume changes in the electrode is obtained from an overall material balance of the solid hase electrode active material + reaction roduct + binder + filler + etc. as t + u = sˆ nf j The local electrode velocity, u, is a vector having the comonents of magnitudes, u x, u y, u z in the x, y, z directions, resectively. Setting the normal velocities to zero at the lanes, x =0,y = 0, z =0 gives the boundary conditions at w =0,u w =0, w = x, y, z 2 This means that i the x velocity of the electrode current collector interface is chosen as reference set to zero ii the y z velocities of the center lanes y = 0 z = 0 are zero, a result of symmetry. The initial condition is set as at t =0, = 0 Exing Eq. using Eq. 2, we get 3

A80 Journal of The Electrochemical Society, 53 A79-A86 2006 Figure. Schematic reresentation of a orous deosition/reciitation electrode before after exansion. The electrode dimensions orosity are functions of time osition. For illustrative uroses, the active material the reaction roduct are shown regularly shaed. The electrode current collector interface is set as x = 0. The center lanes of the electrode in the y z directions are set as y = 0 z = 0, resectively. sˆ t + u j + u = nf 4 at w =0, t + v=x,y,z vw u v v + u sˆ j =, w = x, y, z 5 nf For a differential volume element, d, in the electrode, Eq. 4 means that the rate of volume change due to roduct formation the right side is equal to the total rate of change of orosity the bracketed term on the left side, which is equal to d ln /dt lus the rate of change of volume the u term, which is equal to d ln d/dt. However, Eq. 4 5 are not sufficient to determine the magnitude of each of these terms. In other words, we have two unknowns, u, but only one governing equation, Eq. 4. Rigorously, a force balance equation is needed that relates the orosity velocity to the hysical mechanical roerties of the electrode the cell. However, we resolve this roblem emirically by introducing a swelling coefficient, g, such that Eq. 4 can be slit into sˆ j u = g 6 nf j sˆ + u = g t nf 7 Eq. 5 into sˆ j at w =0, u = g w = x, y, z 8 nf, at w =0, t + v=x,y,z vw u v v sˆ = g j, w = x, y, z 9 nf This means that a fraction, g, of the volume change due to roduct formation goes to change the electrode dimensions the rest goes to change the orosity. For a given volume change due to roduct formation, we now have Eq. 3, 7, 9, to solve for, Eq. 2 6 to solve for u. For a differential volume element, d, the swelling coefficient, g, is defined mathematically as d ln d d ln g d ln d + d ln 0 The swelling coefficient deends on the conditions in the electrode such as the arrangement of articles, the mechanical roerties of the electrode comonents, the header sace available for electrolyte, the ressure exerienced by either face of the electrode. While for a given electrode cell setu these quantities may be constant, g is robably a function of electrode orosity. In an electrode dense with articles i.e., when orosity is low, a greater fraction of volume change due to roduct formation would go to change the electrode dimensions i.e., g is large. For an electrode with high orosity, greater room is available to accommodate the articles, resulting in a greater change in orosity i.e., g is small. Regardless, the actual value or functional form of g for an electrode should be determined exerimentally for a given electrode cell design. The limits on g are 0 ; g = 0 means all the volume change due to roduct formation goes to change the electrode orosity with its dimensions remaining constant, g = means all the volume change due to roduct formation goes to change the dimensions of the electrode with its orosity remaining constant. As mentioned earlier, the limiting case of g = 0 is the more common assumtion used in the literature to model volume change.,4-7 The volume fraction of the electrode s active material, e,is calculated from material balance as e + e u =0 t with the initial condition 0 at t =0, e = e 2 the boundary conditions at w =0, e t + v=x,y,z vw u v e v + e u =0, w = x, y, z 3 Unlike Eq., Eq. need not be slit because u is calculated together with using Eq. 2 6 Eq. 3, 7, 9, resectively. The geometric volume of the electrode is calculated using the relation /t ud 4 = the initial condition at t =0, = 0 5 If exansion occurs equally in all directions, then the changes in the dimensions of the electrode is given by the cube root of the change

Journal of The Electrochemical Society, 53 A79-A86 2006 A8 in electrode volume. However, when exansion occurs differently in different directions, we need to calculate the individual comonents of the velocity to calculate the dimensions of the electrode. In other words, rewriting Eq. 6 in terms of the comonent velocities as u x x + u y y + u z sˆ j = g 6 z nf shows that we have three unknowns, u x, u y, u z, only one governing equation, Eq. 6. To obtain the comonent velocities we introduce further slitting arameters, g x, g y, g z, such that Eq. 6 can be slit into u sˆ x x = g j xg 7 nf u sˆ y y = g j yg 8 nf u sˆ z z = g j zg 9 nf The arameters, g x, g y, g z, determine how much of the electrode s geometric volume change occurs due to change in its dimensions in the x, y, z directions, resectively. Mathematically, they are defined as d ln dx g x 20 d ln d d ln dy g y d ln d 2 d ln dz g z 22 d ln d Thus, we now have three governing equations i.e., Eq. 7-9 the boundary conditions i.e., Eq. 2 to solve for the three comonent velocities, u x, u y, u z. The corresonding electrode dimensions, L x y,z, L y z,x, L z x,y, are obtained as functions of time from the equations dl x = u x x=lx 23 dt dl y dt = u y y=ly 24 dl z = u z z=lz 25 dt with the initial conditions that at t =0,L x = L 0 x, L y = L 0 0 y, L z = L z 26 In the above equations we have introduced four arameters, namely, g, g x, g y, g z. However, adding Eq. 7-9 comaring with Eq. 6 gives g x + g y + g z = 27 Further, in most electrodes the changes in dimensions would occur equally in the y z directions, giving g y = g z = g x 28 2 Thus, in most situations the required number of arameters is reduced to two: g g x. The secial case of uniform reaction current. The model resented above is rigorous alicable for all deosition/ reciitation orous electrodes. However, for illustrative uroses we consider the secial case of an electrode with a uniform initial orosity a uniform reaction current, giving j = I =0 29 30 where I is the total current in ameres alied to or sulied by the electrode. Therefore, j,, the electrode dimensions, L x, L y, L z, are functions of time only. Further, Eq. 30 means that the second terms on the left sides of Eq. 7 9 become zero, therefore, they can be solved indeendently of u. Finally, for a constant g for galvanostatic oeration i.e., for I = constant, Eq. 2, 3, 6, 7, 9, 4 are solved analytically to obtain 0 = + 0 0 t e e 0 = + 0 0 t 0 g 0 g + 0 0 0 = 0 t g 3 32 33 In Eq. 3-33, 0 is a characteristic oerating time of the electrode defined as 0 0 0 I sˆ 34 nf which is the ratio of the initial ore volume to the rate of ore filling. Thus, 0 gives the galvanostatic oerating time i.e., the time taken to fill the ores comletely for the case when there is no change in the geometric volume of the electrode i.e., when g =0. When both the electrode orosity geometric volume change i.e., when g 0, the oerating time,, is calculated from Eq. 3 by setting to zero. Thus 0 = 0 g/ g 0 0 35 The dimensions of the electrode are obtained by solving Eq. 2 7-26 as L x L x 0 = + 0 0 t L y L y 0 = + 0 0 t L z L z 0 = + 0 0 t 0 g x g 0 g y g 0 g z g A + 0 0 A 0 = 0 t g y +g z g = + 0 0 0 t g x g 36 37 38 39 The changes in the ionic electronic resistances of the electrode due to volume change during oeration are given by R i 0 R = L x /L 0 x i A/A 0 / 0.5 40

A82 Journal of The Electrochemical Society, 53 A79-A86 2006 R e 0 R = L x /L 0 x e A/A 0 e / 0 e.5 4 where the ionic electronic conductivities are assumed to be roortional to.5.5 e, resectively. 7 The kinetic resistance of the electrode remains a constant because it is a function of the quantity, a, which is roortional to the constant quantity, e see Eq. 32 33. Intercalation electrodes. In an intercalation electrode, volume changes occur in the solid hase when the roduct of electrochemical reaction intercalates into the active material articles of the electrode. As the articles ex during intercalation, they occuy a greater art of the ore volume at the same time ush each other away. If the volume change of mixing is negligible, the same equations as for deosition/reciitation electrodes are alicable. When volume change in mixing is significant, the governing relationshi between orosity electrode dimensions is derived by observing that the number of articles in the electrodes is conserved as the electrode volume changes. Thus t + u =0 42 where is the volume of an electrode article / is the number of articles er unit volume of the electrode. Exing Eq. 42 we get t + u + u = t + u 43 which means that the volume change in the electrode articles due to intercalation the right side translates into the combined effect of the change in orosity the bracketed term on the left side the change in electrode dimensions the u term. If volume change in mixing is negligible, then the change in article volume is roortional to the local reaction rate. Therefore, Eq. 43 reduces to Eq. 4 with ˆ reresenting the molar volume of ure intercalate. When volume change in mixing is significant, Eq. 43 should be used, wherein the article volume is obtained from material balance of the intercalation rocess occurring inside the article. The rest of the analysis is the same as for deosition/reciitation electrodes excet that here e =. For the secial case of uniform reaction distribution under galvanostatic discharge, the orosity electrode volume are solved analytically as 0 = 0 = L x L x 0 = L y L y 0 = L z L z 0 = g 0 g 0 g 0 x g g 0 y g g 0 z g A A 0 = g 0 y +g z g = g 0 x g 44 45 46 47 48 49 Note that in contrast to deosition/reciitation electrodes, the time of oeration for an intercalation electrode is indeendent of ore Figure 2. A lot of Eq. 35 showing the effect of g the initial orosity, 0, on the oerating time,. The gray line is / 0 =/ g, which is the limit as 0 0. As, / 0 aroaches the y axis. The oint marked shows that for the case of Jain et al., where / 0 =.267 0 = 0.8, g = 0.07. volume therefore, is indeendent of the swelling coefficient. Ionomer-based electrodes. In an ionomer-based electrode such as a carbon ionomer fuel cell catalyst, volume changes occur in the ionomer hase when it absorbs water. As the ionomer volume increases, it occuies a greater volume fraction in the electrode at the same time ushes the active material articles away from each other. Alying a material balance over the ionomer hase gives the governing relationshi between the volume fraction of the ionomer hase, i, the local velocity of the electrode as t ic i + i c i u =0 On exing, Eq. 50 becomes 50 i i + u t i + u = c i c i + u c t i 5 which is of the same form as Eq. 4 or 43. The right side of Eq. 5 gives the volume change in the ionomer hase, where the ionomer concentration, c i, is obtained from a material balance over the absorbed secies. Following the same analysis as for deosition/ reciitation electrodes, i is obtained by solving Eq. 5 along with the initial condition that 0 at t =0, i = i 52 The orosity of the electrode is calculated by noting that = i e. Results Discussion For a given reaction rate distribution, jx,y,z,t, the model resented here can be used to calculate the orosity electrode dimensions as functions of time osition in the electrode. To calculate the reaction rate distribution, the model should be solved simultaneously with charge balances in the ionic electronic hases material balances for the various secies involved. However, for illustrative uroses, here we analyze the effect of volume changes in a deosition/reciitation electrode under conditions of uniform current distribution when the analytical solutions given in Eq. 3-4 aly. The dimensionless oerating time, / 0, given by Eq. 35, is a function of two arameters, g 0, indeendent of g x.in Fig. 2, Eq. 35 is used to lot / 0 vs g for various initial orosities,

Journal of The Electrochemical Society, 53 A79-A86 2006 A83 Figure 3. A lot of Eq. 3 showing the effect of the swelling coefficient, g, on the electrode orosity during oeration. The initial orosity, 0, is 0.4. The electrode orosity is indeendent of the value of g x. Figure 4. A lot of Eq. 32 showing the effect of the swelling coefficient, g, on the volume fraction of the electrode s active material during oeration. The initial orosity, 0, is 0.4. The active material volume fraction is indeendent of the value of g x. 0. By definition, / 0 = when g = 0, i.e., the oerating time aroaches the characteristic oerating time, 0, when there is no change in the geometric volume of the electrode. As g increases, the oerating time increases, aroaching infinity as g. In this limit, the geometric volume change of the electrode is unconstrained discharge continues indefinitely. That is, the volume of the electrode continues to increase to accommodate the roduct formed while the orosity remains constant. For a given g, the dimensionless oerating time, / 0, decreases with decrease in the initial orosity, aroaching the limit / g as the initial orosity aroaches zero. The actual discharge time also decreases with decrease in initial orosity aroaching zero as the initial orosity aroaches zero. When the initial orosity is high i.e., as 0 the oerating time aroaches infinity even for small values of g. The oint marked shows that for the case of Jain et al., where / 0 =.267 0 = 0.8, g = 0.07. Most ractical battery systems would robably reside in the lower left region of Fig. 2. Even in these ractical cases, though, the volume change effects are significant. The swelling coefficient, g, is not an intrinsic roerty of an electrode but rather a design-deendent one. An exing electrode can be designed to have a higher g value by using, for examle, a more elastic binder or by roviding extra sace for exansion. For deosition/reciitation electrodes, the actual value of g is obtained for each design using discharge data cell arameters in conjunction with Fig. 2 or Eq. 35. For examle, Jain et al. observed a low-rate discharge caacity of 6.7 Ah. Based on the initial thickness 0.07 cm initial orosity 0.8 measured rior to cell assembly, a caacity of 3.8 Ah was estimated by Jain et al., assuming that no swelling occurred i.e., g =0. Therefore, / 0 = 6.7/3.8 =.267. Using this value for / 0 with 0 = 0.8, a value for g = 0.07 is obtained from Fig. 2 or Eq. 35. The evolution of orosities, e, with time are functions of g 0 only, as seen from Eq. 3 32. In Fig. 3 4, these equations are lotted for various values of g, keeing 0 = 0.4. Figure 3 shows that when g, the orosity decreases with time reaches zero when t/ 0 = / 0. For 0 = 0.4, Fig. 2 gives the dimensionless oerating times as / 0 =, 2.66, for g = 0, 0.5,, resectively. When g =, the orosity remains constant, therefore, the oerating time is infinitely large. Figure 4 shows the change in the volume fraction of the electrode active material, e, during oeration. When g 0 the geometric volume of the electrode increases causing e to decrease with time because e = constant. When g = 0, the geometric volume remains constant therefore, e also remains constant. The orosity,, the active material volume fraction, e, the geometric volume of the electrode,, are indeendent of g x, as seen from Eq. 3-33. The evolution of L x, A, R i, R e with time are functions of g, 0, as well as g x, as seen from Eq. 36 39-4. In Fig. 5-8, these equations are lotted as functions of time for 0 = 0.4 for various combinations of g g x. In the figures, the black lines are lotted for various values of g, keeing g x = /3, while the gray lines are lotted for various values of g x, keeing g = 0.5. For the case of no volume change i.e., g =0, both the electrode thickness, L x, cross-sectional area, A, remain constant at their initial values. When g 0, both thickness area change with time, with area increasing faster than thickness when g x = /3, as shown by the black lines. For a given g, the electrode thickness area change with time such that their roduct i.e., the electrode volume, remains the same for all values of g x. However, with increase in g x, volume change in the electrode occurs more through thickness change less through area change. In the limit when g x =0, thickness remains constant all the volume change occurs Figure 5. A lot of Eq. 36 showing the effect of the arameters, g g x,on the electrode thickness during oeration. The initial orosity, 0, is 0.4. The black lines are lotted for various values of g, keeing g x = /3. The gray lines are lotted for various values of g x, keeing g = 0.5.

A84 Journal of The Electrochemical Society, 53 A79-A86 2006 Figure 6. A lot of Eq. 39 showing the effect of the arameters, g g x,on the electrode area during oeration. The initial orosity, 0, is 0.4. The black lines are lotted for various values of g, keeing g x = /3. The gray lines are lotted for various values of g x, keeing g = 0.5. Figure 8. A lot of Eq. 4 showing the effect of the arameters, g g x, on the electronic resistance of the electrode during oeration. The initial orosity, 0, is 0.4. The black lines are lotted for various values of g, keeing g x = /3. The gray lines are lotted for various values of g x, keeing g = 0.5. through area change. Conversely, when g x =, area remains constant all the volume change occurs through thickness change. Note that from Eq. 36 the electrode length deends only on the roduct of g g x, causing the black lines g x =/3 gray lines g = 0.5 to overla when the roduct is the same. Similarly, from Eq. 39 the electrode area deends only on the roduct of g g x, causing the black gray lines to overla when their roduct is the same. Figure 7 8 show the effect of volume changes on the ionic electronic resistances, R i R e, of the electrode. As before, the black lines are lotted for various values of g, keeing g x =/3, while the gray lines are lotted for various values of g x, keeing g = 0.5. For g = 0 or 0.5, orosity decreases, consequently, ionic resistance increases, while for g = the ionic resistance decreases even though the orosity remains constant. This is because the resistance deends also on the ratio, L x /A, which decreases because area increases faster than thickness when g x = /3. In contrast, e decreases faster than L x /A, resulting in an increase in the electronic resistance when g 0. When g =0, e, L x, A remain constant, therefore, the electronic resistance also remains constant. Finally, the ionic electronic resistances of the electrode increase with g x because L x /A increases with g x, as seen from Eq. 36 39. Note that if the ionic electronic resistances are equally imortant at the beginning of oeration, then the electrode becomes ionically limited for small values of g electronically limited for large values of g. Further, the end of oeration is generally determined by a cutoff voltage, which deends on the resistances in the electrode. For examle, in the case of a LiSOCl 2 battery cathode of Jain et al., a dro in the electrode voltage of.5 from its initial value is taken to be end of discharge. Assuming that the electrode is ionically limited, this means that for the low-rate discharge, where the initial voltage dro is 0.3, discharge ends when R i /R i 0 = 5. For g =0or 0.5, we see that this occurs well before the ores are comletely filled. In contrast to the ionic electronic resistances, the kinetic resistance of the electrode remains a constant not shown for all values of g g x if the secific surface area of the electrode varies linearly with the volume fraction of the electrode s active material. Figure 7. A lot of Eq. 40 showing the effect of the arameters, g g x,on the ionic resistance of the electrode during oeration. The initial orosity, 0, is 0.4. The black lines are lotted for various values of g, keeing g x =/3. The gray lines are lotted for various values of g x, keeing g = 0.5. Figure 9. A lot of Eq. 3 showing the effect of g on orosity during oeration, when 0 = 0.8 g x =. The gray line shows the method of Jain et al.

Journal of The Electrochemical Society, 53 A79-A86 2006 A85 Figure 0. A lot of Eq. 32 showing the effect of g on volume fraction of the electrode s active material during oeration, when 0 = 0.8 g x =.The gray line shows the method of Jain et al. Figure 2. A lot of Eq. 40 showing the effect of g on the ionic resistance of the electrode during oeration, when 0 = 0.8 g x =. The gray line shows the method of Jain et al. As mentioned in the Model Develoment section, the kinetic resistance deends on the roduct e, which is a constant. As mentioned earlier, g can be obtained by using the exerimentally observed oerating time in Eq. 35. Similarly, g x can be obtained exerimentally from Eq. 36 the knowledge of the change of electrode thickness during oeration. Alternatively, g x can be estimated by fitting Eq. 40 4 to measured resistance during oeration. Figures 9-3 show the effect of volume changes in the electrode for the case when 0 = 0.8 g x =i.e., constant cross-sectional area. The black lines are lotted for g = 0, 0.07, 0.5,, while the gray lines are based on the aroach of Jain et al. They accounted for swelling by using higher values for electrode thickness 0.085 cm initial orosity 0.835 than what was measured rior to cell assembly so as to match the observed caacity i.e., 6.7 Ah. The case of g = 0.07 corresonds to the LiSOCl 2 battery cathode of Jain et al., marked in Fig. 2. While the orosity change with time redicted by Jain et al. lies close to ours Fig. 9, the electrode thickness Fig. 0 active material volume fraction Fig. are qualitatively different. This results in a maximum error of 25% in the ionic electronic resistances Fig. 2 3, consequently, on the electrode voltage. For examle, for the moderate-rate discharge considered by Jain et al., the initial voltage dro is 0.6, which means a maximum error of 0.5 when using their method. At higher rates, this error would roortionally increase because the initial voltage dro directly scales with the discharge rate. Further, larger errors in the redicted voltages would occur for higher values of g because the electrode thickness active material volume fraction change more. The effect of increasing g on the electrode orosity, active material volume fraction, length are qualitatively the same as observed earlier in Fig. 3-6. However, because g x = the electrode thickness increases faster, resulting in the ionic electronic resistances reaching cutoff values much sooner than orosity becoming zero. In contrast to Fig. 7 8, the oerating times based on cutoff voltage in Fig. 2 3 decrease with increase in g if electronically limited, or increase first then decrease with increase in g if ionically limited. Conclusion A mathematical model was develoed to describe volume changes in three dimensions occurring in i deosition/ reciitation, ii intercalation; iii ionomer-based orous elec- Figure. A lot of Eq. 36 showing the effect of g on electrode thickness during oeration, when 0 = 0.8 g x =. The gray line shows the method of Jain et al. Figure 3. A lot of Eq. 4 showing the effect of g on the electronic resistance of the electrode during oeration, when 0 = 0.8 g x =. The gray line shows the method of Jain et al.

A86 Journal of The Electrochemical Society, 53 A79-A86 2006 trodes. The effects of volume changes on the electrode are identified to be twofold, namely, change in electrode dimensions change in orosity. Based on material conservation the governing relationshi between orosity electrode dimensions is derived. Key design-deendent arameters, g g x, are introduced, which determine the individual magnitudes of the changes in orosity electrode dimensions. These arameters should be obtained exerimentally for a given cell design as discussed. For a given reaction rate, the models resented here allow calculation of orosity electrode dimensions as functions of time osition in the electrode. To calculate the reaction rate distribution the model should be solved simultaneously with charge balances in the ionic electronic hases material balances for the various secies involved. For the secial case of galvanostatic oeration with a uniform current distribution, analytical solutions are resented used to illustrate the effect of volume changes. It is shown that the time of ore filling increases steely with increase in g, which, in the case of a battery, means that electrode swelling results in greater discharge caacity. However, the oeration time based on cutoff values of the resistances in the electrode increase or decrease with g deending on the value of g x. The sensitivity of the oerating time the ionic electronic resistances to these arameters demonstrates the imortance of including volume change effects in orous electrode models. The models resented in this work are general form critical additions required to extend the existing orous electrode models to include volume change effects. The University of South Carolina assisted in meeting the ublication costs of this article. List of Symbols a secific surface area of orous electrode, cm 2 /cm 3 A cross-sectional area of orous electrode, cm 2 c i concentration of ionomer in ionomer hase, mol/cm 3 F Faraday s constant, 96,487 C/mol g swelling coefficient introduced in Eq. 6 7 g x, g y, g z slitting arameters introduced in Eq. 7-9 I total alied current, A j local volumetric electrochemical reaction rate, A/cm 3 L x, L y, L z dimensions of electrode in x, y, z directions, cm n number of electron transfers in electrochemical reaction R e electronic resistance of orous electrode, R i ionic resistance of orous electrode, s stoichiometric coefficient of the roduct in electrochemical reaction t time, s u local velocity vector in the electrode, cm/s u x, u y, u z magnitudes of the comonents of u in the x, y, z directions, cm/s v defined when used total electrode volume, cm 3 ˆ molar volume of reaction roduct, cm 3 /mol volume of intercalation article, cm 3 w defined when used x, y, z osition in the electrode, cm Greek orosity e volume fraction of electrode active material i volume fraction of ionomer hase oerating time or time taken to fill ores comletely, s 0 oerating time or time taken to fill ores comletely when g =0,s Suerscrit 0 initial References. M. Jain, G. Nagasubramanian, R. G. Jungst, J. W. Weidner, J. Electrochem. Soc., 46, 4023 999. 2. J. Yang, Y. Takeda, Q. Li, N. Imanishi, O. Yamamoto, J. Power Sources, 90, 64 2000. 3. P. Choi, N. H. Dalani, R. Datta, J. Electrochem. Soc., 52, E23 2005. 4. R. C. Alkire, E. A. Grens II, C. W. Tobias, J. Electrochem. Soc., 6, 328 969. 5. J. S. Dunning, D. N. Bennion, J. Newman, J. Electrochem. Soc., 8, 25 97. 6. J. S. Dunning, D. N. Bennion, J. Newman, J. Electrochem. Soc., 20, 906 973. 7. J. Newman K. E. Thomas-Alyea, Electrochemical Systems, 3rd ed.,. 536 537, 523, John Wiley & Sons, New York 2004. 8. G. G. Botte, Abstract 34, The Electrochemical Society Meeting Abstracts, ol. 2003-2 Orlo, FL, Oct 2 6, 2003. 9. J. Christensen J. Newman, Abstract 6, The Electrochemical Society the Electrochemical Society of Jaan Meeting Abstracts, ol. 2004-2 Honolulu, HI, Oct 3 8, 2004.