Multiscale Computational Model for Particle Size Evolution during


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B: Liquids, Chemical and Dynamical Processes in Solution, Spectroscopy in Solution

Multiscale Computational Model for Particle Size Evolution during Coprecipitation of Li-Ion Battery Cathode Precursors Pallab Barai, Zhange Feng, Hiroki Kondo, and Venkat Srinivasan J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b12004 • Publication Date (Web): 19 Mar 2019 Downloaded from http://pubs.acs.org on March 26, 2019

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The Journal of Physical Chemistry

Multiscale Computational Model for Particle Size Evolution during Coprecipitation of Li-ion Battery Cathode Precursors

Pallab Barai, Zhange Feng, Hiroki Kondo, Venkat Srinivasan* Argonne National Laboratory, Lemont, IL, USA 60439

Revised version submitted to Journal of Physical Chemistry B March 2019

*Corresponding author: [email protected]

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Abstract Next generation lithium ion batteries require higher energy and power density, which can be achieved by tailoring the cathode particle morphology, such as, particle size, size distribution, and internal porosity. All these morphological features are determined during the cathode synthesis process, which consists of two steps, i) coprecipitation, and ii) calcination. Transition metal hydroxide precursors are synthesized during the coprecipitation process, whereas, their oxidation and lithiation occur during calcination. The size and size distribution of crystalline primary and aggregated secondary particles, and their internal porosity are determined during coprecipitation. Operating conditions of the chemical reactor, such as, solution pH, ammonia concentration and stirring speed control the final morphological features. Here, a multiscale computational model has been developed to capture the nucleation and growth of crystalline primary particles, and their aggregation into secondary transition metal hydroxide precursor particles. The simulations indicate that increasing solution pH and decreasing ammonia concentration leads to smaller size of the secondary particles. A phase-map has been developed that can help identify the synthesis conditions needed for a specified particle size and size distribution.

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Introduction Lithium ion batteries are being extensively used in the electronics and automobile industry due to its high energy and power density and enhanced cycle life as compared to other existing rechargeable battery chemistries (such as, lead-acid and nickel-metalhydride)1. However, increasing demand of longer run times for electronic devices, and the need to eliminate range anxiety associated with electric vehicles, has driven the research community to develop next generation lithium ion batteries with higher energy density2. Commercially available lithium ion batteries use graphite as the anode, and lithium-cobalt-oxide (LCO) or layered nickel-manganese-cobalt (NMC) based cathode materials3. As demonstrated in Figure 1, cross sectional image of a NMC 532 cathode active particle reveals the presence of sub-micron sized crystalline primary particle aggregates4. The larger secondary particles span over several micrometers in diameter. The specific capacity and energy density of cathode active particles depend not only on the chemical composition, but also on the size and size distribution of the secondary particles, internal porosity and primary particle size. These morphological features of the active particles are usually determined during the synthesis process5. In most of the industrially adopted techniques, firstly, the NMC (111, 532, 622, 811, etc.) hydroxide precursors are prepared from transition metal sulfates through the coprecipitation process5-6. However, there exist other processes where transition metal carbonates or oxalates are coprecipitated from metal sulfate or nitrate based salts 7. Next, the precursors are mixed with lithium-hydroxide (LiOH) or lithium-carbonate (Li2CO3) and calcined at

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high temperatures (~10000C) in oxygen atmosphere to oxidize, lithiate and sinter the transition metal precursors and form lithium transition metal oxides (Li(NMC)O2)5-6.

Figure: 1. Focused Ion Beam – Scanning Electron Microscopy (FIB-SEM) image of the cross section of an NMC 532 secondary particle usually used as the active material for cathodes in lithium ion batteries (adopted from Gilbert et al., JES 164 (2) A389 – A399 (2017))4. Note that the FIB-SEM has been conducted on an oxide secondary particle obtained after coprecipitation and calcination.

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To maximize the specific capacity and energy density of lithium ion battery cathodes, it is important to thoroughly understand the different steps involved in the preparation of the cathode materials, and its impact on the particle morphology6b, 8. At present, the synthesis of lithium ion battery cathode materials is largely based on conducting multiple experiments and developing empirical relations that approximately correlate the synthesis conditions (e.g., coprecipitation solution pH, coprecipitation solution ammonia content, calcination temperature, excess lithium during calcination, etc.) with cell performance (e.g., capacity, rate performance, etc.)

6, 7b, 8-9.

A

detailed scientific understanding of the different physical phenomena that occur during synthesis is clearly missing in the existing literature. Optimization of various cathode manufacturing techniques require running the same experiment several times with different operating parameters, which may require significant investment in time and cost 10.

Developing computational models capable of simulating the various physical

phenomena that occur during synthesis can significantly accelerate the rate of development of new materials by eliminating the costly trial-and-error approach. In this manuscript, we focus on understanding the different physical phenomena that occurs inside a chemical reactor during the coprecipitation process. We report the first developed multiscale computational model capable of predicting the particle size and size distribution as a function of operating parameters, such as, solution pH, and ammonia content. This paper represents the first step in developing a computational framework to predict the synthesis of hydroxide precursors, where the model is capable of capturing the variation in reactant concentration as well as evolution of both primary and secondary

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particles. A computational model to predict the calcination process will be adopted as our next step.

Methodology Coprecipitation of transition metal hydroxide precursors is usually conducted in continuous stirred tank reactors (CSTR). In the present context, we focus at the initial few hours of operation of the reactor, where inflow of reactants are occurring, but outflow has not been initiated, which can also be characterized as a batch (or semi-batch) reactor. Different phenomena that occur within the reacting solution, starting from chemical reactions to physical nucleation, growth and aggregation of particles, will be described in this section. The computational model developed to capture these physico-chemical phenomena will also be introduced. A list of equations, which were used in the computational model, has been provided within Table: 1, and further details regarding these equations is given in the Supplementary section (all equation numbers given in Table 1 starts with L). A schematic diagram of a standard batch reactor is shown in Figure 2, along with some other reactants that play major roles in the coprecipitation of transition metal hydroxides6b. For the coprecipitation process, desired transition metal sulfates (MSO4, M=Ni,Mn,Co) mixed in stoichiometric amount, are pumped into the reactor at a fixed rate (shown in Figure 2 at the left top corner). The reactor is maintained at constant pH (~ 9.0-12.0) through the addition of sodium hydroxide (NaOH) (shown in the right side of Figure 2). High pH is desirable to ensure the precipitation of metal hydroxides. Even though the solubility product of transition metal hydroxides are very small in magnitude

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(Ksp ~ 10-15), square of the concentration of the hydroxide anions (𝐾𝑠𝑝 = [𝑀2 + ][𝑂𝐻 ― ]

2

)

make it impossible to precipitate at pH smaller than 76b. Constant concentration of ammonia is maintained within the chemical reactor, which acts as a chelating agent and prevents formation of impurity phases (see left bottom side of Figure 2)10b. Inert nitrogen gas is continuously bubbled into the reactor to remove the dissolved oxygen (not shown in schematic diagram). Excess air or oxygen within the solution can cause unwanted oxidation of the hydroxides and form transition metal oxides or oxy-hydroxides6a. The reactor is continuously stirred (at around 1000 rpm) to ensure good mixing of the reactants (shown at the center of the batch reactor in Figure 2). The stirring process leads to formation of turbulence within the reactor and affects the particle aggregation process11.

Figure: 2. Simplified schematic diagram of a batch reactor used for conducting the coprecipitation process. A constant magnitude of pH is maintained inside the reactor by pumping in sodium hydroxide according to the requirements. For precipitating transition metal hydroxide particles, transition metal sulfate solution of concentration 2mol/l is slowly added into the reactor at a fixed rate. Here, in MSO4, M stands for the transition metal, which can be Ni, or Co, or Mn, or Ni(1/3)Mn(1/3)Co(1/3), depending on what is being precipitated. Ammonium hydroxide is added at a fixed rate to maintain a constant concentration of ammonia within the reactor. During operation, the reactor is stirred at a fixed rate to ensure good mixing of the reactants and the products.

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Formation of transition metal hydroxide precursors from the metal sulfate solution involves multiple physico-chemical phenomena. Due to the high solubility product of the transition metal sulfates, they readily dissolve in water12. If the pH of this solution is increased, by addition of NaOH, transition metal hydroxides may precipitate. But there is a chance of formation of impurity phases, such that individual Ni(OH)2, or Mn(OH)2, or Co(OH)2 may form, instead of (NMC)(OH)210b, 13. To prevent the formation of individual phases, ammonia is added to the solution as a chelating agent, which forms metal ammonia complex with the transition metals and minimizes the precipitation of individual phases10b,

14

(see Eq. (L1) in Table: 1). It has been argued by several

researchers that for the formation of spherical and uniformly distributed cathode precursors, it is necessary that the hydroxyl anions react with the metal ammonia complex for the precipitation of transition metal hydroxides6b, 10b (see Eq. (L2) in Table: 1). Two different pathways for this reaction have been proposed: a) The hydroxyl anions slowly react with the metal ammonia complex to form the metal hydroxide precipitates10b. b) There exist equilibrium between the metal hydroxide and metal ammonia complex, which effectively increases the solubility of the hydroxide precipitates6b. Both the mechanisms point at the importance of having sufficient amount of ammonia in the reactor. Details of the chemical reactions have been provided in the Supplementary Information section. In general, it is hypothesized that three different physical phenomena play major roles within the chemical reactors during the coprecipitation process, namely, nucleation,

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growth and aggregation15. In the present research effort, we have developed a computational methodology that can capture these physical phenomena11, 16: a) Nucleation of transition metal hydroxides, which occurs in the nanometer range, is described using the Classical Nucleation Theory. b) Growth of the nuclei, which leads to the formation of submicron sized primary particles, is described using a continuum-based growth model. c) Aggregation of sufficiently large primary particles, which leads to the formation of micron sized secondary particles, is captured using a Monte Carlo scheme. It is evident that the solution pH and amount of ammonia present within the solution significantly affects formation of primary and secondary particles6b, 8. The developed computational model has been used to elucidate the impact of these reactants on the primary and secondary particle morphology. Comparisons with experimental results have also been conducted to validate the applicability of the developed computational model. Nucleation process: At higher pH, as the product of metal ammonia complex and hydroxyl anion concentration exceeds the solubility product of the metal hydroxides, it forms a supersaturated solution17. Extent of super-saturation is estimated through super2

saturation ratio, 𝑆, defined as, 𝑆 = 𝑐𝑀𝑁𝐻3 ∙ 𝑐𝑂𝐻 𝐾𝑠𝑝, where, 𝑐𝑀𝑁𝐻3 indicates bulk concentration of metal ammonia complex, 𝑐𝑂𝐻 denotes the bulk concentration of hydroxyl anions, and 𝐾𝑠𝑝 stands for the solubility product. It has been assumed that all the metal cations within the solution react with ammonia and form metal-ammonia complex, and co-precipitation of the transition metal hydroxide occurs only through the slow reaction between metal ammonia complex and hydroxyl anions10b, 14. Please note that, this assumption may not be valid at higher values of pH (>11.5), where solubility of

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metal ammonia complex decreases significantly, and coprecipitation of transition metal hydroxides may occur through the reaction between metal cations and hydroxyl anions6b. More detailed analysis and comparison between the relative amount of metal hydroxide precipitates formed through the reaction with metal cations and metal ammonia complex has been elaborated within the Supplementary section (see Figure S2). Inside this super-saturated solution, transition metal hydroxides start to precipitate in the form of small nuclei, and the extra energy required for the formation of these nuclei comes from the continuous agitation of the reactor fluid11, 18. Based on Classical Nucleation Theory (CNT), formation of nuclei can be explained from the minimization of combined surface and bulk energies, where both contribute to the total Gibbs free energy (see Eq. (L5) in Table: 1). The critical nucleus size (𝑟 ∗ ), beyond which the particles are stable and grow in size to minimize Gibbs free energy, is given as19: 𝑟 ∗ = (3𝛾𝑉) (𝑅𝑔𝑇ln 𝑆)

(1)

where, 𝛾 indicates the surface energy of the metal hydroxide precipitate, 𝑉 denotes partial molar volume of the precipitate, 𝑆 indicates the super-saturation ratio, 𝑅𝑔 is the universal gas constant, and 𝑇 indicates absolute temperature (in Kelvin scale). By using appropriate parameters, the initial radii of the transition metal hydroxide nuclei are estimated to be around 5 angstroms. Details of the mathematical expression used to calculate the rate of nucleation (𝐽) is provided in the Supplementary Information section. Growth of primary particles: Once the nuclei are formed, it is surrounded by supersaturated solution of transition metal hydroxides. Hence, precipitation must occur on top of these nuclei for minimization of the bulk Gibbs free energy20. Since the primary particles are assumed to be spherical in shape, transport of reactants from the bulk

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solution to the particle surface has been captured by solving the diffusion equation in spherical coordinates, which can be written as follows: ∂𝑐𝑀𝑁𝐻3 ∂𝑡

∂2𝑐𝑀𝑁𝐻3

= 𝐷𝑀𝑁𝐻3

∂𝑟2

+

2𝐷𝑀𝑁𝐻3∂𝑐𝑀𝑁𝐻3 ∂𝑟

𝑟

.

(2)

The boundary condition is defined as the flux at the particle surface (𝑟 = 𝑅𝑖(𝑡)) is equivalent to the growth rate of the primary particle: 𝑑𝑅𝑖(𝑡)

𝑐𝑀(𝑂𝐻)2

𝑑𝑡

∂𝑐𝑀𝑁𝐻3

= ― 𝐷𝑀𝑁𝐻3

∂𝑟

, at 𝑟 = 𝑅𝑖(𝑡)

(3)

and, concentration of the reactant far away from the particle surface (𝑟 = 𝑅0) reaches that of the bulk value: 𝑐𝑀𝑁𝐻3(𝑟 = 𝑅0,𝑡) = 𝑐𝑀𝑁𝐻3, at 𝑟 = 𝑅0.

(4)

Here, 𝑡 is time, 𝑟 denotes the distance from the particle surface, 𝑐𝑀𝑁𝐻3(𝑟,𝑡) is the local concentration of metal ammonia complex that varies with position and time, and 𝐷𝑀𝑁𝐻3 indicates the diffusion coefficient of metal ammonia complex within the reacting solution. The primary particles are assumed to be of size 𝑅𝑖(𝑡), which increases with time. Outer boundary of the computational domain is denoted by 𝑅0 and spans to a sufficiently long distance where bulk concentration of the metal ammonia complex can be approximated. Concentration inside the precipitated particle is denoted as 𝑐𝑀(𝑂𝐻)2. This precipitation on top of the nuclei allows its growth and formation of crystalline primary particles21. Growth rate of the primary particles are determined from the amount of super-saturation and precipitation rate of the transition metal hydroxides16b, 20, 22: 𝑑𝑅𝑖(𝑡) 𝑑𝑡

(

𝑘𝑟𝑒𝑓 𝑐𝑀𝑁𝐻3 ―

=

(𝐾𝑠𝑝 𝑐2𝑂𝐻))

𝑐𝑀(𝑂𝐻)2

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(5)

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where, 𝑘𝑟𝑒𝑓 denotes the rate of precipitation reaction, and the other terms have already been explained earlier. Details of how to solve the diffusion equation and estimate the growth of primary particles have been described in the Supplementary section. The growth mechanism can be either kinetic rate controlled or species diffusion controlled 16b. The particle size increase linearly with time for rate limited growth, and is proportional to square root of time for species transport limited growth22. Depending on the crystal structure of the transition metal hydroxides, the growth may be preferred in certain crystal orientations that correspond to increase in minimum energy surfaces23. However, crystal facet dependent growth of primary particles has not been taken into consideration in the present model. It should be noted that the growth specified here corresponds to that of the primary or crystalline particles. Diffusion of primary particles: In a basic solution (with pH>7), the surfaces of the transition metal hydroxide particles become charged due to proton abstraction24 (see Eq. (L7) in Table: 1). This surface charge induces an electrostatic repulsion among crystalline primary particles that prevents them from aggregating11. In addition, continuous stirring of the reactor solution is also thought to prevent agglomeration for nanometer sized primary particles25. In the present computational methodology, it has been assumed that the nanometer sized primary particles, with diameter < 150 nm, do not agglomerate26. Hence, prior to aggregation, the primary particles move randomly within the reacting solution due to turbulent convection and agitation8. A diffusion process has been assumed to control this random movement of the primary particles (see Eq. (L8) in Table: 1), which depends on the diffusivity and concentration of the reactants (elaborated in the Supplementary section). Also, note that the magnitude of aggregation threshold (assumed

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to be 150 nm in the present case) may have some impact on the size of primary and secondary particles, which has been clearly elaborated in the Supplementary section. It is worth mentioning that the screening length of the proton abstraction induced surface charge (see Eq. (L7) in Table: 1) may not be constant, and vary according to the concentration of reactants within the liquid. For lower concentration of the reactants, which is observed during the initial phase of the coprecipitation process, the screening length is large, and the surface charge can be visualized from a long distance. However, after conducting coprecipitation for couple of hours, concentration of reactants increases, and the screening length for the surface charge decreases substantially. Hence, the primary particles should not experience the repulsion due to surface charge and may come closer to each other. However, for the aggregation to occur, the particles must attach to each other. For higher concentration within the reacting liquid, the screening length decreases, but the surface charge still remains. When the primary particles try to agglomerate, the surface repulsion force keeps them separate, and the particles can grow in size due to precipitation from the solution. At higher concentration of the reactants, decrease in screening length may have an impact on the growth of the primary particles. The second boundary condition, applicable to concentrations at far away distances, (see Eq. (4)), gets affected by the variation in screening length. Aggregation process: Due to precipitation induced growth, the primary particles increase in size20. As the size of the primary particles grows, the number of collision between different primary particles increases25. As the size of the primary particles keep increasing, intensity of two different attractive forces grows, enhancing the propensity of these primary particles to agglomerate11, 27. These forces are:

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a) Van der Waals force, which is directly proportional to the size of the particle. b) Turbulent agglomeration, which is proportional to the surface area of the particle. Under favorable conditions, sufficiently large primary particles aggregate to form secondary particles. These secondary particles can also agglomerate with the surrounding primary or secondary particles11. Based on these discussions it is evident that the possibility of the primary particles to form aggregates increase with their size, which leads to the formation and growth of the secondary particles11. Formation of secondary particles has been captured using the Monte Carlo (MC) technique, which tries to minimize the total energy of the system28. Energy between the primary particle and the liquid solution has been assumed to consist of two different components29. a) Energy associated with agitation of the primary particles, which is assumed to be proportional to the stirring speed. This particular form of energy dominates for the smaller particles and tries to keep them floating within the liquid solution (see Eq. (L9) in Table: 1). b) Surface energy between the primary particle and the surrounding liquid solution, which is proportional to the surface area of the particles. This energy term leads to aggregation of primary particles and minimization of the solid-liquid interface. This agglomeration energy is dominant on larger particles (see Eq. (L10) in Table: 1). Competition between these two different forms of energy helps to capture the formation and growth of secondary particles within the reacting solution. More details of the particle aggregation process have been provided within the Supplementary section.

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The above-mentioned nucleation, growth and agglomeration steps have been implemented on a square lattice domain of size 400x400. Length of each lattice site has been assumed to be 300 nm. Irrespective of its size (nanometer sized nuclei or submicron sized primary particles), each primary particle can occupy only one lattice site, and vice versa. Temperature of the reacting fluid has been assumed to remain constant at 333K. The set of equations and a list of parameters used in the simulation have been provided in Table 1 (main article) and Table S2 (Supplementary section), respectively. Table: 1. List of equations used in development of the computational model. Name

Num ber

Equation

Metalammonia 2+ 𝑀2 + + 𝑛𝑁𝐻3→[𝑀(𝑁𝐻3)𝑛] (L1) complex formation Metalhydroxide [𝑀(𝑁𝐻3)𝑛]2 + + 2𝑂𝐻 ― ⇌𝑀(𝑂𝐻)2 + 𝑛𝑁𝐻3 (L2) precipitati on reaction Metal cation 𝑐𝑀2 + = [𝑀2 + ] + [𝑀(𝑂𝐻)2] + [𝑀(𝑁𝐻3)2 + ] + [𝑀(𝑁𝐻3)22 + ] + [𝑀(𝑁𝐻3)23 + ] mass (L3) + [𝑀(𝑁𝐻3)24 + ] + [𝑀(𝑁𝐻3)25 + ] + [𝑀(𝑁𝐻3)26 + ] balance equation Ammonia 𝑐𝑁𝐻3 mass = [𝑁𝐻3] + [𝑁𝐻4𝑂𝐻] + [𝑀(𝑁𝐻3)2 + ] + 2[𝑀(𝑁𝐻3)22 + ] +(L4) 3[𝑀(𝑁𝐻3)23 + ] + 4 balance [𝑀(𝑁𝐻3)24 + ] + 5[𝑀(𝑁𝐻3)25 + ] + 6[𝑀(𝑁𝐻3)26 + ] equation ―16𝜋𝛾3𝑉2 Rate of 𝐽 = 𝐽0exp (L5) nucleation 3𝑘𝐵𝑅2𝑔𝑇3(ln 𝑆)2

(

Number density of nuclei Proton abstraction reaction Diffusivity of primary particles

)

𝑁𝐷(𝑡 + 𝛥𝑡) = 𝑁𝐷(𝑡) + (𝐽.𝛥𝑡)

(L6)

𝑀(𝑂𝐻)2 + 𝑂𝐻 ― ⇌𝑀𝑂𝑂𝐻 ― + 𝐻2𝑂

(L7)

𝐷𝑝 = 𝐷𝑝,0 ∙

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𝐷𝑀𝑁𝐻3𝑐𝑁𝐻3 𝑐𝑂𝐻

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(L8)

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Agitation energy Agglomer ation energy

Page 16 of 48

𝐸𝑎𝑔𝑖 = ― 𝐴𝑓 ∙ 𝛺

(L9)

𝐸𝑎𝑔𝑔𝑙𝑜 = 4𝜋(𝑅𝑖(𝑡))2 ∙ 𝛾

(L10)

Results and Discussion Through implementation of the chemical equilibrium of various species, nucleation of hydroxide precipitates, and growth/aggregation of primary particles, successful simulation of the formation of transition metal hydroxide secondary particles has been attempted. The list of equations has been provided in Table 1. The list of parameters used in the present simulation has been given in Tables S1 and S2 within the Supplementary section. The equilibrium constants and the solubility products used in the computational procedure have been determined based on the weighted average of the different transition metal cations present within the reacting solution, because in a mixed solution for higher concentration of the metal cations, they precipitate at a fixed rate

30.

Following are three different features, regarding the morphology of particles, observed in general coprecipitation experiments: a) Bimodal particle size distribution within the reacting solution8 b) Decrease in secondary particle size with increasing solution pH10c c) Increase in secondary particle size with increasing ammonia content10c In this study, ability of the developed computational model to capture these experimentally observed features have been investigated. Following these directions, in the present manuscript, optimum magnitudes of pH and ammonia concentration required for the coprecipitation of uniformly sized particles, will be estimated.

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Concentration of metal-ammonia complex with changing pH: Since precipitation of the transition metal hydroxides occurs through the reaction of metal ammonia complex and hydroxyl anions (see Eq. (L2) in Table: 1), it is very important to determine the amount of metal ammonia complex that exist within the solution as a function of pH and ammonia content6b,

7b.

Figure 3 demonstrates the concentrations of different metal

ammonia complex as obtained by solving the mass balance and equilibrium equations provided in Eqs. (S3), (S4) and Table S1, respectively (shown in the Supplementary Information section).

Figure: 3. Concentration of metal ammonia complex formed within the reacting solution during coprecipitation conducted at different pH levels. Magnitudes of these metalammonia complexes have been obtained by solving the equilibrium and mass balance equations for dissolved metal-ions and ammonia within the solution (adopted from van Bommel and Dahn Chem. Mater. 21 1500 – 1503 (2009))6b. It is evident that higher solubility of the metal ammonia complex is observed only at the intermediate pH range (4.0 < pH < 12.0). In the pH range where coprecipitation reactions are usually conducted (9.0 < pH < 12.0), the solubility of metal ammonia complex decreases monotonically with increasing pH.

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For this particular calculation, bulk concentration of transition metal and ammonia has been assumed to be 2.0mol/L and 1.0mol/L, respectively6b. It is evident from the figure that higher solubility of the metal ammonia complex is obtained at intermediate ranges of pH (4.0 < pH < 12.0). Coprecipitation is usually conducted at higher pH (9.0 < pH < 12.0), to ensure the presence of sufficient amount of hydroxyl anions within the solution, required for successful precipitation of transition metal hydroxides (also highlighted in Figure 3). The numbers of ammonia molecules that coordinate with metal cations at different pH is also shown in Figure 3. It should be noted that at the pH range of interest (9.0 < pH < 12.0), usually larger numbers of ammonia molecules (n > 3), coordinate with the metal cations to form the metal ammonia complex. At even higher magnitudes of pH (pH > 12.0), solubility of the metal ammonia complex decreases significantly, and cannot ensure uniform precipitation of the transition metal hydroxides. Hence, conventional coprecipitation reactions are not conducted at high pH. Growth of primary particles: As transition metal start to flow into the reacting solution, the metal cations coordinate with the available ammonia molecules to form transition metal ammonia complex (as shown in Eq. (L1) in Table: 1)6b, 10b. High pH of the solution maintained within the reactor provides the required hydroxyl anions for coprecipitation to occur. Transition metal hydroxides start to precipitate according to the chemical reaction given in Eq. (L2) (see Table: 1). For successful precipitation, firstly small nuclei of the metal hydroxides should form. Number density of the nuclei can be determined by combining Eqs. (L5) and (L6) (see Table: 1)15. Once the initial nuclei are developed, they continue to grow according to the mathematical expression given in Eq. (5). Change in concentration of the metal ammonia complex in the vicinity of the

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growing particle is determined by solving Eq. (2), along with the boundary conditions given in Eqs. (3) and (4). The smaller primary particles randomly move around due to the stirring induced agitations29b. Once their size exceeds a certain threshold, the primary particles start to aggregate, and form the secondary particles11. After agglomeration of the primary particles, they lose contact with the reacting liquid, and their growth stops. Other primary particles, which are either located on the surface of the secondary particle, or extremely small to form aggregates, remain in contact with the reacting fluid, and continue to grow in size. Evolution of the average primary particle diameter with time has been demonstrated in Figure 4(a). Before the formation of any secondary particle, only primary particles exist within the reacting solution. Continuous growth of each of these particles leads to an increase in the average primary particle diameter (Regime I in Figure 4(a) observed at smaller times). Linear increase of primary particle diameter with time reveals a rate controlled growth of the crystalline phase

 R t  ~ t  20. i

As the

agglomeration starts to occur, secondary particles appear within the solution. Primary particles trapped inside the secondary ones lose access to the reacting fluid and stop growing in size. Hence the average primary particle diameter ceases to increase with time, or grow with a very slow rate (Regime II in Figure 4(a) observed at larger times).

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Figure: 4. (a) Computationally predicted evolution of primary particle size with time during the coprecipitation process conducted at different magnitudes of pH. Eventual saturation of the primary particle size is observed due to aggregation and loss of contact with the reacting solution. (b), (c) Primary particle size distribution obtained at pH 9.0 and 12.0, respectively, measured at the end of coprecipitation process. (d), (e) Distribution of primary and secondary particles within the computational domain for pH 9.0 and 12.0, respectively. Relatively slower growth of the primary particles is observed at higher pH, which is attributed to the reduced solubility of the metal ammonia complex. Increase in number density of the nuclei and decrease in growth rate, results in smaller size of the primary particles at higher pH (compare the number density of particles in (b) and (c)).

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It is also evident from Figure 4(a) that with increasing solution pH, growth rate of the primary particle decreases. This can be attributed to the reduction in concentration of metal ammonia complex observed at high pH values (see Figure 3), which leads to a significant drop in the rate of precipitation (demonstrated in Eq. (1)). Even though the concentration of metal ammonia complex decrease at higher values of pH (see Figure 3), the super-saturation ratio  S  increases with pH due to the quadratic dependence on the concentration of the hydroxyl anions. Larger super-saturation lead to enhanced number density of the nuclei at high pH (see Eqs. (L5) and (L6) in Table: 1). This is evident from Figures 4(b) and 4(c), which demonstrates the distribution of primary particles. Significantly large number of small sized primary particles (smaller than 100nm) observed at pH=12.0, clearly supports the claim. Hence, at high pH, the primary particles are located closer to each other, which increases their propensity to aggregate. Quick agglomeration, and loss of contact with the reacting fluid, limits the size of the primary particles at higher values of pH. Hence, the average primary particle size decreases with the increase in solution pH. This mechanism can also be understood from Figures 4(d) and 4(e), where the distribution of particles within the computational domain has been demonstrated for pH 9.0 and 12.0, respectively. Impact of solution pH on the secondary particle size will be discussed later. Growth of secondary particles: As the size of the primary particles increase, their propensity to form aggregates also rise11. Agglomeration of primary particles and formation of the secondary particles have been captured using a lattice based Monte Carlo procedure31. The black solid line in Figure 5(a) demonstrates the increase in

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average secondary particle diameter, as observed in a batch reactor operating under pH=11.0 and ammonia concentration cNH3=1.0mol/L.

Figure: 5. (a) Time evolution of secondary particle size obtained during coprecipitation reaction conducted at pH=11.0 and ammonia concentration cNH3=1.0mol/L. Very close correlation between the particle size evolution curve (black solid line) and the square root of time curve (red dashed line) clearly indicates that in the computational model, evolution of the secondary particle size is a diffusion controlled process. The error bars (in blue) indicate the first standard deviation of particle size distribution at that particular time. Decrease in size distribution at times greater than 6 hours can be attributed to the presence of Ostwald ripening effect. (b), (c), (d), (e) Distribution of primary and secondary particles within the computational domain after 1 hour, 2 hours, 4 hours and 12 hours of operation.

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The secondary particle size varies as directly proportional to the square root of time

(𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟~(𝑡𝑖𝑚𝑒)1 2), which has been demonstrated by the red dashed line in Figure 5(a)20, 22. Since, prior to aggregation, all the primary particles diffuse randomly within the solution, the diffusion process itself controls their arrival at the surface of a secondary particle. Hence, diffusion controlled growth of the secondary particles is indeed reasonable. Such diffusion controlled growth of the secondary particle has also been observed experimentally7b,

9

(see Figure S6(A) for a direct experimental evidence of

secondary particle growth adopted from Ref. 9). The blue vertical error-bars (shown in Figure 5(a)) indicate the particle size distribution, which have been determined by calculating the first standard deviation of the entire population of secondary particle. For small times (1hr-2hr), since the size of the particles is small, their size distribution is also small. At intermediate times (4hr-6hr), larger error-bars indicate non-uniform size distribution of the secondary particles. However, after long time operation (8hr-12hr), the particle size distribution becomes much narrow, indicating a relatively uniform distribution of particle sizes. The aggregation phenomena occur in order to minimize energy of the entire solution32. Uniform size of the spherical particles lead to minimum surface area, and hence minimizes the surface energy, which can be attained only after operation for a long time. Formation of secondary particles with narrow size distribution happens through agglomeration of smaller secondary particles with larger ones, which has been characterized as the Ostwald ripening phenomena6b. Even though re-dissolution and reprecipitation of the reactants have not been modeled explicitly, Ostwald ripening has been captured through the Monte Carlo scheme where smaller particles coalesce with

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larger ones in order to minimize the overall surface energy. The two regimes of increasing size distribution due to growth of secondary particles, and decrease in size distribution due to Ostwald ripening, have been clearly demonstrated in Figure 5(a). Distribution of secondary particles within the computational domain shown in Figures 5(b), 5(c), 5(d) and 5(e) corresponds to that obtained after 1hr, 2hr, 4hr and 12hr, respectively. This helps to visualize the particle sizes and size distributions as it evolves with time within the computational domain. Bimodal particle size distribution within reactor: Size distribution of “isolated” particles within the computational domain has been demonstrated in Figure 6. To clarify the nomenclature of “isolated”, if a bunch of primary particles agglomerate and form a secondary particle, the entire aggregate is considered to be a single “isolated” particle. Extremely small primary particles floating within the solution have also been considered as isolated ones. Bimodal size distribution of these isolated particles has been observed after 3 hours, 6 hours, 9 hours and 12 hours (shown in Figures 6(a), 6(b), 6(c) and 6(d), respectively). In the bimodal distribution, the smaller particles are the primary ones, which keep floating within the solution, whereas the larger particles correspond to the secondary aggregates. Some experimental research claim that there may be a bimodal distribution within the secondary particles as well8, 33, however it is not reflected in the size distribution diagram. The computational simulation assumed constant pH=11.0 and ammonia concentration of 1.0mol/L within the batch reactor. With increasing time, more primary particles agglomerate with the secondary ones, which leads to an increase in average secondary particle diameter. Size of the primary particles remains approximately constant. But their number density decreases with time, because of growth and

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aggregation with existing secondary particles. Similar bimodal distributions have also been observed during the particle size analysis of solutions obtained from chemical reactors8-9 (see Figure S6(B) for an example of experimentally observed bimodal particle size distribution within chemical reactors, adopted from Ref. 9). This positive correlation with the experimental observation acts as a qualitative validation for the developed computational model.

Figure: 6. During the coprecipitation process at pH = 11.0 and ammonia concentration cNH3 = 1.0mol/L, the developed model predicts bimodal particle size distribution within the computational domain. Bimodal particle size distribution within the reactor after: (a) 3 hours, (b) 6 hours, (c) 9 hours and (d) 12 hours have been demonstrated here. The smaller sized particles (size less than 0.5μm) corresponds to the primary particles, whereas, the larger particles (size greater than 1μm) indicate the secondary ones. As time increases, due to aggregation of the primary particles, size of the secondary particles keeps increasing. Similar bimodal particle size distribution has also been observed in regular coprecipitation experiments8-9.

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Primary particle distribution within the secondary particle: Distribution of various primary particles within the secondary one has been clearly demonstrated in Figure 7. The developed computational procedure predicted the diameter of the secondary particle to be approximately 7μm. The simulation has been conducted at pH=11.0 and ammonia concentration cNH3=1.6mol/L. One interesting feature is that the primary particles located near the surface demonstrate smaller size than the ones near the center. From computational perspective, the particles located near the surface nucleate later than the ones near the center. Hence, these primary particles close to the surface do not have enough time to grow in size. The focused-ion-beam-scanning-electronmicroscopy (FIB-SEM) image of the cathode active particle reveals relatively large sized primary particles near the surface (see Figure 1)4. However, it should be noted that the FIB-SEM image has been captured at the cross-section of a lithiated transition metal oxide particle (Li(NMC)1/3O2), which may contain larger particles at the surface due to high temperature calcination step 6a. It should be noted that the distribution of primary particles inside the secondary one, depends on how the reactor is operated. For the present simulation of a batch reactor, where transition metal solution has been added continuously into the reacting solution, the smaller primary particles exist near the surface and larger ones lie at the center. For a different kind of batch reactor, where all the reactants are mixed at the very beginning of the reaction34, the distribution of the primary particles, inside the secondary one, may be very different from that observed here.

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Figure: 7. Computationally predicted secondary active particle obtained from a coprecipitation process at pH=11.0 and cNH3=1.6mol/L, which contains primary particles of various sizes. Note that the demonstrated distribution of primary particles within the secondary particle has been obtained right after the coprecipitation process.

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Impact of solution pH on secondary particle size: Since the solution pH significantly impacts the growth, aggregation and diffusion of primary particles, pH of the solution will definitely affect the size of secondary particles. The black solid line in Figure 8(a) demonstrates the dependence of secondary particle size on solution pH, which has been obtained after running the coprecipitation simulation for 12 hours. Ammonia content of the solution has been maintained constant at 1.6mol/L. To compare the computational predictions with experimental observations, average size of the secondary particles obtained from coprecipitation experiments have been indicated by red squares in Figure 8(a)10c. Higher concentration of ammonia has been maintained during the simulation in order to create equivalent conditions as reported in the experimental literature. As observed in both experiment and computation, increasing the solution pH leads to a decrease in secondary particle size. Because of the proton abstraction reaction mentioned in Eq. (L7) (see Table: 1), negative electrostatic charge evolves at the surface of the primary particles24. As the solution pH increases, enhanced strength of this electrostatic repulsion among adjacent primary particles leads to a decrease in their diffusion coefficient, which eventually limits the average size of the secondary particle. The black error-bars indicate the particle size distribution, which has been estimated as the first standard deviation of the secondary particle diameters. The size distribution is relatively higher for solution pH smaller than 11.0, but decreases drastically for higher magnitudes of pH (>11.0). One possibility can be that the decreasing particle size at higher pH effectively renders a smaller size distribution. It

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should also be noted that, large solution pH leads to higher nucleation density and smaller diffusivity of the primary particles (see Eq. (L5) and (L8) in Table: 1).

Figure: 8. (a) Variation in secondary particle size obtained at the end of coprecipitation process conducted at different magnitudes of pH. The concentration of ammonia has been kept constant for all the reactions (cNH3 = 1.6mol/L). The black line indicates the computational prediction along with error bars demonstrating the first standard deviation in the particle size distribution. The red squares indicate average secondary particle sizes observed during coprecipitation experiments (adopted from Noh and Cho, JES 160 (1) A105 – A111 (2013))10c. (b), (c), (d), (e) The predicted particle size distribution within the computational domain demonstrated for pH values of 9.0, 10.0, 11.0 and 12.0, respectively. It is evident that at higher magnitudes of pH (~ 12.0), even though the secondary particle size decreases, its density increases significantly. This can be attributed to the higher density of nuclei, enhanced number of primary particles and reduced propensity of aggregation at larger magnitudes of pH.

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Due to lower diffusivity, primary particles from a large distance cannot come and aggregate with existing secondary particles. Hence, at high pH, the secondary particles get sufficient time to reorganize (through surface-diffusion) and form uniformly distributed spherical particles, which effectively leads to smaller size distributions. Figures 8(b), 8(c), 8(d) and 8(e) demonstrates the distribution of both primary and secondary particles within the computational domain after coprecipitation conducted at pH 9.0, 10.0, 11.0 and 12.0, respectively. The number density of secondary particles within the computational domain clearly increases with increasing solution pH. At pH=11.0, magnitude of the average particle size is around 8μm, which is slightly larger than that reported in Figure 5 (obtained at ammonia concentration of 1.0mol/L). This can potentially be attributed to the higher ammonia content (1.6mol/L) adopted for comparison with experimental results10c. Impact of ammonia concentration on the solution pH will be discussed next. Effect of ammonia on secondary particle size: Ammonia content of the reacting solution determines the amount of metal ammonia complex, which directly impacts the nucleation density (Eq. (S8)), primary particle growth rate (Eq. (5)) and their diffusivity (Eq. (L8) in Table: 1). Hence, it is expected that the ammonia concentration will have a direct impact on size of the secondary particles. Black solid line in Figure 9(a) demonstrates the rise in secondary particle diameter with increasing ammonia content (during operation at pH=11.0). The vertical error-bars denote first standard deviation of the particle size distributions, which shows a narrow distribution of secondary particles at lower concentrations of ammonia. As demonstrated in Eq. (L8) (see Table: 1), the diffusivity of primary particles increases with ammonia content. Inside reactors with

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large ammonia concentration, primary particles located far away from the secondary particle can diffuse and form aggregates because of their higher diffusivity. Hence, in high ammonia solution, size of the secondary particles keeps changing due to agglomeration between adjacent secondary particles, till the end of the reaction. Figures 9(b), 9(c), 9(d) and 9(e) demonstrates the distribution of primary and secondary particles within the computational domain after conducting coprecipitation for 12 hours under ammonia concentration of 0.8mol/L, 1.6mol/L, 3.0mol/L and 5.0mol/L, respectively. Agglomeration of multiple secondary particles, and formation of tertiary particles can be clearly observed from the images of the computational domain obtained after coprecipitation with high ammonia concentration (see Figures 9(d) and 9(e)). These larger tertiary particles act as the secondary particle at higher magnitudes of ammonia content, which increase both size and size distribution of the secondary aggregates. Existence of large tertiary particles has also been observed in some other experimental studies8. For the purpose of comparison with experimental results, average secondary particle diameter obtained at various levels of ammonia concentration has been demonstrated in Figure 9(a) by red squares10c. The computational predictions diverge significantly from the experimental data. However, the qualitative trend of increasing secondary particle size with increasing ammonia content has been successfully captured. The mismatch between experiment and computation can be attributed to the enhanced mobility of the primary particles at large concentrations of ammonia, which cannot be captured by the present model due to limited size of the computational domain. Reason behind the decrease in particle size at ammonia concentration of 3.0 mol/L is not clear,

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and cannot be captured by the present computational technique. Increasing the computational domain size may lead to more accurate estimation of the secondary particle size.

Figure: 9. (a) Variation in secondary particle size with ammonia concentration (cNH3) during coprecipitation at pH = 11.0. The black curve demonstrates computationally predicted average particle size with error bars indicating the first standard deviation of the size distribution. Enhanced propensity of aggregation among the primary particles at higher ammonia content leads to larger secondary particles, but the standard deviation also increases resulting in higher uncertainty for the secondary particle size. Experimentally observed variations in secondary particle size with ammonia content have been shown by the red squares (adopted from Noh and Cho, JES 160 (1) A105 – A111 (2013))10c. (b), (c), (d), (e) At the end of coprecipitation, distribution of primary and secondary particles within the computational domain at ammonia concentrations of 0.8mol/L, 1.6mol/L, 3.0mol/L and 5.0mol/L, respectively. With increasing ammonia content, a combination of enhanced metal-ammonia complex formation and higher diffusivity of the primary particles lead to the formation of larger secondary particles.

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Note that the experimental data shown in Figure 8, 9 and S6, all have been collected from continuous stirred tank reactors (CSTRs), which may have different residence times. Since, reaction for longer times may lead to larger particles, the computational efforts have been devoted to understanding the qualitative variation in particle sizes, and not their exact magnitudes. Also, a paragraph has been added within the Supplementary section describing the applicability of the present analysis at the initial (transient) phase of coprecipitation in a CSTR. Also, none of the experimental studies adopted from the literature showed results regarding the particle size distribution. Hence, no comparison of the particle size distribution could be conducted between computational and experimental results. Phase-map demonstrating optimum solution pH and ammonia content: Finally, a phase map has been developed between the ammonia concentration and solution pH to demonstrate how these operating parameters impact the average secondary particle size. The yellowish portion in the contour plot shown in Figure 10 depicts larger secondary particles, whereas the greenish region corresponds to smaller sized aggregates. It is evident that high ammonia content and low solution pH leads to larger secondary particles, due to enhanced formation of metal ammonia complex (see Figure 3) and higher diffusivity of the primary particles (see Eq. (L8) in Table: 1). However, it should be noted that along with particle size, the size distribution of secondary particles also plays a major role in determining the electrochemical performance of an active material6a, 8. Narrow particle size distributions lead to better performance because of reduced surface area and less unwanted side reactions. To estimate the particle size distribution, the first standard deviation has also been depicted

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in Figure 10 using small circles. Light blue corresponds to narrow size distribution, whereas, purple indicates wide particle size variation. Both the particle sizes and size distributions, reported in Figure 10, have been obtained after running the simulation for 12 hours. It should be noted that small sized particles always experience lower magnitudes of standard deviation. Hence, the extremely low size distribution observed at high pH and low ammonia content (right bottom corner of the contour plot) is not useful, because the particle sizes are also extremely small there. As predicted by the present computational approach, two regions have been highlighted in the contour plot with red circles, which denotes the optimum operating conditions for precipitating large and small sized secondary particles: a) Larger particles of size around 8μm and relatively narrow size distribution are obtained at pH~10.5 and cNH3~1.0mol/L. b) Smaller particles of size 4μm and uniform size distribution can be obtained at pH~11.5 and cNH3~1.5mol/L. These optimum operating conditions are not unique, and depend significantly on the computational techniques adopted in the present analysis. Realistic primary particles are never spherical in shape35. Their disk-type shape can affect the growth and aggregation mechanism, which has the potential to alter the optimum operating conditions6a. Also, adopting non-spherical particles may allow the computational model to capture the evolution of internal porosity within the secondary particles. Please note that the present computational technique cannot capture the variation in phase concentration within the particles, and assumes all the particles to be in NMC 111 phase. Also, tap density of the computationally predicted secondary particles have not been reported, because the

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internal porosities are not being captured, and lack of algorithms that can estimate the random packing of particles while preserving its shape.

Figure: 10. Phase map between solution pH and ammonia content demonstrating the average particle sizes and corresponding first standard deviation under the assumption of Gaussian particle size distributions. All the data shown here have been adopted from computational model. The “greenish” region indicates smaller sized secondary particles, whereas, the “yellowish” region stands for larger particles. Light blue colored dots demonstrate smaller value of standard deviation, whereas, purple dots indicate larger magnitude of standard deviation for the secondary particle size distribution. Optimum operating conditions, in terms of pH and ammonia content, for precipitating relatively larger (Dpart ~ 8μm) and smaller (Dpart ~ 4μm) sized secondary active particles, have also been highlighted within the figure (by red circles).

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Conclusion A simplified multiscale computational methodology has been developed that is capable of successfully predicting the evolution of transition metal hydroxide particles during the coprecipitation process. The computational framework implemented in this study takes into consideration the chemical equilibrium of relevant species, nucleation of metal hydroxides, growth of crystalline primary particles, and finally their aggregation to form the secondary particles. As soon as the metal sulfate solution comes in contact with ammonia, it forms a metal ammonia complex. Precipitation of transition metal hydroxides happens through the reaction between hydroxyl anions and metal ammonia complex6b, 10b. As the size of growing primary particles exceed a certain threshold, they start to agglomerate and form the secondary particles11. Both size and size distribution of the cathode secondary particles have the capability to impact the performance and cycle life of the cell6a. During the coprecipitation process, pH and ammonia content of the reacting solution significantly affects the secondary particle morphology6b. The developed computational technique reveals that the growth of crystalline primary particles is rate limited (see the growth controlled domain in Figure 4(a)), whereas, the growth of secondary particles is a diffusion limited process (see Figure 5(a)). Also, both primary and secondary particles coexist simultaneously within the reacting solution8. The growth rate of primary and secondary particles has been estimated as 3.5 nm/min and 0.05 μm/min, respectively, from the present computational analysis, which compares well with experimental observations of 5 nm/min and 0.15 μm/min 26. It has also been observed that size of the secondary particles rise with increasing ammonia content and decreasing solution pH. This can be attributed to the larger number density of

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nuclei and enhanced primary particle diffusivity at low solution pH and high ammonia concentration (see Eqs. (L5) and (L8) in Table: 1). Finally, a phase map has been developed in Figure 10, between solution pH and ammonia concentration, that tries to estimate an optimum operating condition for obtaining desired average particle sizes with narrow size distributions. However, it should be noted that there exist several approximations within the adopted computational procedure, which are far from experimentally observed scenarios. Certain physical features that needs to be accounted for to obtain a more accurate estimation of the species concentration and particle size distribution will be discussed here: a) In a coprecipitation process, sodium hydroxide is added to maintain constant pH of the reacting solution6b. The equations solved for determining the chemical equilibrium neglected the presence of sodium hydroxide within the solution. Since, the volume of added NaOH can impact the total volume of solution within the reactor, addition of NaOH must be incorporated within the computational framework. b) In the present research, coprecipitation of metal hydroxides has been assumed to occur through the reaction between metal ammonia complex and hydroxyl anions. However, at high pH (pH > 11.5), since all the transition metal does not react with ammonia to form metal ammonia complex, it is possible to precipitate metal hydroxides through direct reaction between metal cation and hydroxyl anions (see Figure S2 in the Supplementary section for a more detailed analysis). Formation

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of metal hydroxides from metal ammonia complex as well as metal cations, during coprecipitation at high pH, must be taken into account in future studies. c) Shape of the primary particles has been assumed to be spherical in the present context. However, SEM images reveal a disk-like shape of the transition metal hydroxide

primary

particles

prior

to

calcination35.

Hence,

appropriate

modifications should be adopted in the computational framework to successfully elucidate the evolution of internal porosity within the secondary particles. d) In the present study, a certain number has been assumed as the critical primary particle size beyond which it starts to aggregate26. The accurate size can be estimated from in-situ ultra-small-angle-x-ray-scattering (USAXS) experiments36. It may also be possible to develop a computational model that predicts the critical primary particle size based on the solution pH, ammonia content and stirring speed. e) The present version of the computational model does not capture the variation in internal porosity of the secondary particles. This leads to poor estimation of the tap density of coprecipitated particles at higher pH (> 11.5). Proper modifications of the computational scheme must be conducted that is capable of capturing the evolution of internal porosity within secondary particles. Some of them have also been summarized in the Supplementary Information section. Such developments and modifications can definitely help to provide more accurate estimations of secondary particle sizes and their size distributions. Predictions regarding internal porosities may lead to more accurate computational estimation of tap density, which is usually considered to be a very important measure of the particle compactness.

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Supporting Information The following information are provided within the Supporting Information document: 1. Details of the computational procedure 2. Calculations regarding: a. The amount of metal hydroxide precipitate from metal ammonia complex and metal cations b. Tap density estimation c. Effect of variation in agglomeration threshold d. Transient to steady state mode in a CSTR e. Experimentally observed particle growth rate and size distribution (adopted from literature)

Acknowledgements This work has been conducted at Argonne National Laboratory (ANL) and funded by the Roll-To-Roll (R2R) program organized by the Advanced Manufacturing Office (AMO) of the U.S. Department of Energy (DOE) under Contract No. DE-AC0206CH11357. The authors would also like to acknowledge the valuable discussions with Dr. YoungHo Shin, Dr. Ozgenur Kahvecioglu Feridun, Dr. Albert Lipson and other scientists at the Material Engineering Research Facility (MERF) located within ANL.

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List of Symbols Roman 𝐴𝑓

proportionality constant for the agitation energy

𝑐𝑀𝑁𝐻3 concentration of metal ammonia complex in bulk 𝑐𝑀𝑁𝐻3 local concentration of metal ammonia complex around primary particle 𝑐𝑁𝐻3

bulk concentration of ammonia within the reactor

𝑐𝑂𝐻

concentration of hydroxyl anions in bulk

𝑐𝑀(𝑂𝐻)2 concentration of the transition metal hydroxide precipitate 𝐷𝑀𝑁𝐻3 diffusivity of metal ammonia complex within reactant solution 𝐷𝑝

diffusivity of primary particles

𝐷𝑝,0

proportionality constant for the diffusivity of primary particles

𝐸𝑎𝑔𝑔𝑙𝑜 surface energy between primary particle and liquid 𝐸𝑎𝑔𝑖

agitation energy between particle and liquid

𝐽

rate of nucleation

𝐽0

nucleation constant

𝐾𝑠𝑝

solubility product of transition metal hydroxide species

𝑘𝐵

Boltzmann’s constant

𝑘𝑟𝑒𝑓

rate of precipitation reaction

𝑀𝑊𝑀(𝑂𝐻)2

molecular weight of the transition metal hydroxide precipitate 40

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𝑁𝐷

number density of nuclei

𝑅𝑔

universal gas constant

𝑅𝑖

radius of the primary particle

𝑅0

distance from the center of the primary particle where bulk concentration can be assumed

𝑟

radial distance around the primary particle

𝑟∗

critical radius of nucleation

𝑆

super-saturation ratio of the reactants

𝑇

temperature in Kelvin scale

𝑡

time

𝑉

partial molar volume of the transition metal hydroxide precipitate

𝛥𝑡

incremental time

𝛾

surface energy density

Greek

𝜌𝑀(𝑂𝐻)2 𝛺

density of the transition metal hydroxide precipitate

stirring speed

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References (1) Blomgren, G. E., The Development and Future of Lithium Ion Batteries. J Electrochem Soc 2017, 164, A5019-A5025. (2) Myung, S. T.; Maglia, F.; Park, K. J.; Yoon, C. S.; Lamp, P.; Kim, S. J.; Sun, Y. K., Nickel-Rich Layered Cathode Materials for Automotive Lithium-Ion Batteries: Achievements and Perspectives. Acs Energy Lett 2017, 2, 196-223. (3) Nitta, N.; Wu, F. X.; Lee, J. T.; Yushin, G., Li-ion battery materials: present and future. Mater Today 2015, 18, 252-264. (4) Gilbert, J. A.; Shkrob, I. A.; Abraham, D. P., Transition Metal Dissolution, Ion Migration, Electrocatalytic Reduction and Capacity Loss in Lithium-Ion Full Cells. J Electrochem Soc 2017, 164, A389-A399. (5) Zhou, F.; Zhao, X. M.; van Bommel, A.; Rowe, A. W.; Dahn, J. R., Coprecipitation Synthesis of NixMn1-x(OH)(2) Mixed Hydroxides. Chem Mater 2010, 22, 1015-1021. (6) (a) van Bommel, A.; Dahn, J. R., Synthesis of Spherical and Dense Particles of the Pure Hydroxide Phase Ni1/3Mn1/3Co1/3(OH)(2). J Electrochem Soc 2009, 156, A362-A365; (b) van Bommel, A.; Dahn, J. R., Analysis of the Growth Mechanism of Coprecipitated Spherical and Dense Nickel, Manganese, and Cobalt-Containing Hydroxides in the Presence of Aqueous Ammonia. Chem Mater 2009, 21, 15001503. (7) (a) Wang, D.; Belharouak, I.; Zhou, G.; Amine, K., Synthesis of Lithium and Manganese-Rich Cathode Materials via an Oxalate Co-Precipitation Method. J Electrochem Soc 2013, 160, A3108 - A3112; (b) Wang, D. P.; Belharouak, I.; Koenig, G. M.; Zhou, G. W.; Amine, K., Growth mechanism of Ni0.3Mn0.7CO3 precursor for high capacity Li-ion battery cathodes. J Mater Chem 2011, 21, 9290-9295; (c) Zhao, X.; Zhou, F.; Dahn, J. R., Phases Formed in Al-Doped Ni1/3Mn1/3Co1/3(OH)2 Prepared by Coprecipitation: Formation of Layered Double Hydroxide. J Electrochem Soc 2008, 155, A642 - A647. (8) Wang, D. P.; Belharouak, I.; Ortega, L. H.; Zhang, X. F.; Xu, R.; Zhou, D. H.; Zhou, G. W.; Amine, K., Synthesis of high capacity cathodes for lithium-ion batteries by morphology-tailored hydroxide co-precipitation. J Power Sources 2015, 274, 451457. (9) Shin, Y. H.; Feridun, O. K.; Krumdick, G. Scaling Up High-Energy Cathode Materials for Electric Vehicles. http://www.sigmaaldrich.com/technicaldocuments/articles/materials-science/scaling-up-high-energy-cathodematerials.html (accessed 03/12/2019). (10) (a) Kim, M. H.; Shin, H. S.; Shin, D.; Sun, Y. K., Synthesis and electrochemical properties of Li[Ni0.8Co0.1Mn0.1]O-2 and Li[Ni0.8Co0.2]O-2 via co-precipitation. J

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Power Sources 2006, 159, 1328-1333; (b) Lee, M. H.; Kang, Y.; Myung, S. T.; Sun, Y. K., Synthetic optimization of Li[Ni1/3Co1/3Mn1/3]O-2 via co-precipitation. Electrochim Acta 2004, 50, 939-948; (c) Noh, M.; Cho, J., Optimized Synthetic Conditions of LiNi0.5Co0.2Mn0.3O2 Cathode Materials for High Rate Lithium Batteries via Co-Precipitation Method. Journal of the Electrochemical Society 2013, 160, A105-A111. (11) Allen, E.; Smith, P.; Henshaw, J. A Review of Particle Agglomeration; AEAT/R/PSEG/0398; AEA Technology Engineering Services, Inc: Prepared for US Department of Energy, 2001. (12) Hem, J. D. Chemical equilibria and rates of manganese oxidation; Washington, 1963. (13) Schwarz, J. A.; Contescu, C.; Contescu, A., Methods for Preparation of Catalytic Materials. Chem Rev 1995, 95, 477-510. (14) Cho, J., LiNi0.74Co0.26-xMgxO2 cathode material for a Li-ion cell. Chem Mater 2000, 12, 3089-3094. (15) Leubner, I. H., Particle nucleation and growth models. Curr Opin Colloid In 2000, 5, 151-159. (16) (a) Kashchiev, D.; van Rosmalen, G. M., Review: Nucleation in solutions revisited. Cryst Res Technol 2003, 38, 555-574; (b) Zener, C., Theory of Growth of Spherical Precipitates from Solid Solution. Journal of Applied Physics 1949, 20, 950 953. (17) Elimelech, M.; Gregory, J.; Jia, X.; Williams, R. A., Particle Deposition and Aggregation: Measurement Modeling and Simulation. Butterworth-Heinemann: Woburn, MA, USA, 1995. (18) (a) Karthika, S.; Radhakrishnan, T. K.; Kalaichelvi, P., A Review of Classical and Nonclassical Nucleation Theories. Cryst Growth Des 2016, 16, 6663-6681; (b) Vetter, T.; Iggland, M.; Ochsenbein, D. R.; Hanseler, F. S.; Mazzotti, M., Modeling Nucleation, Growth, and Ostwald Ripening in Crystallization Processes: A Comparison between Population Balance and Kinetic Rate Equation. Cryst Growth Des 2013, 13, 4890-4905. (19) Mullin, J. W., Crystallization. 4th ed.; Butterworth-Heinemann: Oxford ; Boston, 2001; p xv, 594 p. (20) Cao, G., Nanostructures & nanomaterials : synthesis, properties & applications. Imperial College Press: London ; Hackensack, NJ, 2004; p xiv, 433 p. (21) Thanh, N. T. K.; Maclean, N.; Mahiddine, S., Mechanisms of Nucleation and Growth of Nanoparticles in Solution. Chem Rev 2014, 114, 7610-7630. (22) Bombaˇc, D. Atomistic Simulations of Precipitation Kinetics in Multicomponent Interstitial/Substitutional Alloys. University of Ljubljana, 2012. (23) Barmparis, G. D.; Lodziana, Z.; Lopez, N.; Remediakis, I. N., Nanoparticle shapes by using Wulff constructions and first-principles calculations. Beilstein J Nanotech 2015, 6, 361-368. (24) Degen, A.; Kosec, M., Effect of pH and impurities on the surface charge of zinc oxide in aqueous solution. J Eur Ceram Soc 2000, 20, 667-673. (25) (a) Gielen, B.; Jordens, J.; Thomassen, L. C. J.; Braeken, L.; Van Gerven, T., Agglomeration Control during Ultrasonic Crystallization of an Active Pharmaceutical Ingredient. Crystals 2017, 7; (b) Mumtaz, H. S.; Hounslow, M. J.; Seaton, N. A.; 43

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Paterson, W. R., Orthokinetic aggregation during precipitation: A computational model for calcium oxalate monohydrate. Chem Eng Res Des 1997, 75, 152-159. (26) Feng, Z.; Barai, P.; Gim, J.; Yuan, K.; Wu, Y.; Xie, Y.; Liu, Y.; Srinivasan, V., In situ monitoring of the growth of nickel, manganese, and cobalt hydroxide precursors during co-precipitation synthesis of Li-ion cathode materials. Journal of the Electrochemical Society 2018, 165, A3077 - A3083. (27) (a) Bhattacharjee, S.; Elimelech, M.; Borkovec, M., DLVO interaction between colloidal particles: Beyond Derjaguin's approximation. Croat Chem Acta 1998, 71, 883-903; (b) Trefalt, G.; Borkovec, M. Overview of DLVO Theory. http://www.colloid.ch/dlvo (accessed 03/12/2019). (28) Bombac, D.; Kugler, G. In Precipitation in Alloys: A Kinetic Monte Carlo And Class Model Study, 19th Annual International Conference on Composites or Nano Engineering, Shanghai, China, Shanghai, China, 2011. (29) (a) Jung, H.; Lee, B.; Jun, Y. S., Structural Match of Heterogeneously Nucleated Mn(OH)(2)(s) Nanoparticles on Quartz under Various pH Conditions. Langmuir 2016, 32, 10735-10743; (b) Pathria, R. K.; Beale, P. D., Statistical Mechanics. Third ed.; Elsevier: Boston, MA, USA, 1996. (30) Dong, H. X.; Koenig, G. M., Compositional control of precipitate precursors for lithium-ion battery active materials: role of solution equilibrium and precipitation rate. J Mater Chem A 2017, 5, 13785-13798. (31) Anderson, M. P.; Srolovitz, D. J.; Grest, G. S.; Sahni, P. S., Computer-Simulation of Grain-Growth .1. Kinetics. Acta Metall Mater 1984, 32, 783-791. (32) Qin, X. G.; Liu, G. Q., Grain growth simulation based on Potts model with different parameters. Mater Sci Forum 2005, 475-479, 3173-3176. (33) Wang, D. P.; Belharouak, I.; Zhou, G. W.; Amine, K., Nanoarchitecture MultiStructural Cathode Materials for High Capacity Lithium Batteries. Adv Funct Mater 2013, 23, 1070-1075. (34) Robinson, J. P.; Koenig, G. M., Tuning solution chemistry for morphology control of lithium-ion battery precursor particles. Powder Technol 2015, 284, 225230. (35) Garcia, J. C.; Bareno, J.; Yan, J. H.; Chen, G. Y.; Hauser, A.; Croy, J. R.; Iddir, H., Surface Structure, Morphology, and Stability of Li(Ni1/3Mn1/3Co1/3)O-2 Cathode Material. J Phys Chem C 2017, 121, 8290-8299. (36) Peyronel, F.; Ilavsky, J.; Mazzanti, G.; Marangoni, A. G.; Pink, D. A., Edible oil structures at low and intermediate concentrations. II. Ultra-small angle X-ray scattering of in situ tristearin solids in triolein. Journal of Applied Physics 2013, 114.

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List of Figures Figure: 1. Focused Ion Beam – Scanning Electron Microscopy (FIB-SEM) image of the cross section of an NMC 532 secondary particle usually used as the active material for cathodes in lithium ion batteries (adopted from Gilbert et al., JES 164 (2) A389 – A399 (2017))4. Note that the FIB-SEM has been conducted on an oxide secondary particle obtained after coprecipitation and calcination. Figure: 2. Simplified schematic diagram of a batch reactor used for conducting the coprecipitation process. A constant magnitude of pH is maintained inside the reactor by pumping in sodium hydroxide according to the requirements. For precipitating transition metal hydroxide particles, transition metal sulfate solution of concentration 2mol/l is slowly added into the reactor at a fixed rate. Here, in MSO4, M stands for the transition metal, which can be Ni, or Co, or Mn, or Ni(1/3)Mn(1/3)Co(1/3), depending on what is being precipitated. Ammonium hydroxide is added at a fixed rate to maintain a constant concentration of ammonia within the reactor. During operation, the reactor is stirred at a fixed rate to ensure good mixing of the reactants and the products. Concentrations of different reactants shown here are representative values, and may not have direct correlation with that used in real experiments. Figure: 3. Concentration of metal ammonia complex formed within the reacting solution during coprecipitation conducted at different pH levels. Magnitudes of these metalammonia complexes have been obtained by solving the equilibrium and mass balance equations for dissolved metal-ions and ammonia within the solution (adopted from van Bommel and Dahn Chem. Mater. 21 1500 – 1503 (2009))6b. It is evident that higher solubility of the metal ammonia complex is observed only at the intermediate pH range (4.0 < pH < 12.0). In the pH range where coprecipitation reactions are usually conducted (9.0 < pH < 12.0), the solubility of metal ammonia complex decreases monotonically with increasing pH. Figure: 4. Computationally predicted evolution of primary particle size with time during the coprecipitation process conducted at different magnitudes of pH. Eventual saturation of the primary particle size is observed due to aggregation and loss of contact with the reacting solution. Relatively slower growth of the primary particles is observed at higher pH, which is attributed to the reduced solubility of the metal ammonia complex. Increase in number density of the nuclei, and decrease in the propensity of aggregation at higher pH results in smaller size of the primary particles. Figure: 5. Time evolution of secondary particle size obtained during coprecipitation reaction conducted at pH=11.0 and ammonia concentration cNH3=1.0mol/l. Very close correlation between the particle size evolution curve (black solid line) and the square root

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of time curve (red dashed line) clearly indicates that in the computational model, evolution of the secondary particle size is a diffusion controlled process. The error bars (in blue) indicate the first standard deviation of particle size distribution at that particular time. Distribution of primary and secondary particles within the computational domain after 1 hour, 2 hours, 4 hours and 12 hours of operation have also been demonstrated in the figure. Figure: 6. During the coprecipitation process at pH = 11.0 and ammonia concentration cNH3 = 1.0mol/l, the developed model predicts bimodal particle size distribution within the computational domain. Bimodal particle size distribution within the reactor after: (a) 3 hours, (b) 6 hours, (c) 9 hours and (d) 12 hours have been demonstrated here. The smaller sized particles (size less than 0.5μm) corresponds to the primary particles, whereas, the larger particles (size greater than 1μm) indicate the secondary ones. As time increases, due to aggregation of the primary particles, size of the secondary particles keeps increasing. Similar bimodal particle size distribution has also been observed in regular coprecipitation experiments8. Figure: 7. Computationally predicted secondary active particle obtained from a coprecipitation process at pH=11.0 and cNH3=1.6mol/l, which contains primary particles of various sizes. Note that the demonstrated distribution of primary particles within the secondary particle has been obtained right after the coprecipitation process. Figure: 8. Variation in secondary particle size obtained at the end of coprecipitation process conducted at different magnitudes of pH. The concentration of ammonia has been kept constant for all the reactions (cNH3 = 1.6mol/l). The black line indicates the computational prediction along with error bars demonstrating the first standard deviation in the particle size distribution. The red squares indicate average secondary particle sizes observed during coprecipitation experiments (adopted from Noh and Cho, JES 160 (1) A105 – A111 (2013))10c. The predicted secondary particle size distribution within the computational domain has also been demonstrated for pH values of 9.0, 10.0, 11.0 and 12.0. It is evident that at higher magnitudes of pH (~ 12.0), even though the secondary particle size decreases, its density increases significantly. This can be attributed to the higher density of nuclei, enhanced number of primary particles and reduced propensity of aggregation at larger magnitudes of pH. Figure: 9. Variation in secondary particle size with ammonia concentration (cNH3) during coprecipitation at pH = 11.0. The black curve demonstrates computationally predicted average particle size with error bars indicating the first standard deviation of the size distribution. Enhanced propensity of aggregation among the primary particles at higher ammonia content leads to larger secondary particles, but the standard deviation also increases resulting in higher uncertainty for the secondary particle size. Experimentally observed variations in secondary particle size with ammonia content have been shown by the red squares (adopted from Noh and Cho, JES 160 (1) A105 – A111 (2013))10c. Distribution of secondary particles within the computational domains has also been depicted by the inset images.

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Figure: 10. Phase map between solution pH and ammonia content demonstrating the average particle sizes and corresponding first standard deviation under the assumption of Gaussian particle size distributions. All the data shown here have been adopted from computational model. The “greenish” region indicates smaller sized secondary particles, whereas, the “yellowish” region stands for larger particles. Light blue colored dot demonstrates smaller value of standard deviation, whereas, pink dots indicate larger magnitude of standard deviation for the secondary particle size distribution. Optimum operating conditions, in terms of pH and ammonia content, for precipitating relatively larger (Dpart ~ 8μm) and smaller (Dpart ~ 4μm) sized secondary active particles, have also been highlighted within the figure (by red circles).

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