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# How To Use the Secondary Pore Network Analytical Diffusion Model (Text Version)

This is a text version of a video about how to use the Secondary Pore Network Analytical Diffusion Model for Lithium-Ion Battery.

Hello, my name is François Usseglio-Viretta and I'm a battery scientist at the National Renewable Energy Laboratory, or NREL. I am the developer of this quite simple yet interesting MATLAB code, and I will explain in this video what it is about and how to use it.

The code runs with the MATLAB live editor environment, which allows you to hide code lines while leaving visible text, equations, figures and results, allowing a user-friendly experience.

Before talking about the model itself, now that you can click on this button on the right here to see the code if you want to understand how it works, or on this button here to hide the code, if you simply want to use the model as end-user, or for presenting results.

For this demonstration I would hide the code and I would also use a full-screen option. You can navigate the model by clicking on the hyperlink of the table of contents table of contents, or you can simply scroll up and down.

First at all. This model has been published in the following article enabling fast charging of lithium-ion batteries through secondary- /dual- pore network: part one, analytical diffusion model. The authors are myself, Weijie Mai, Andrew Colclasure, Marca Doeff and Eongyu Yi from Lawrence Berkeley National Laboratory, and Kandler Smith. Please quote this article if you share this model or some results obtained with it.

Before using the model for the first time it is good to have a little piece of background. You can find such information in the background section of the model.

To make it short:  lithium ion batteries typically experience performance degradation when used under a first charge, especially if they are thick, which is a particle issue for automotive application, as first charge allows to charge quickly an electric vehicle while thick electrodes allows to reduce the battery cost as the amount of electrode chemically insert materials, such as the current collectors, is reduced.

One reason that explains the performance degradation is a transport limitation of the lithium ions within the electrolyte all along the electrode thickness, from the separator to the current collector, and vice-versa.

Limited ionic transport within the electrode implied partial utilization of the electrode volume, and of the active material, and thus eventually a reduction of the capacity. In addition, it can trigger earlier degradation as the front of the electrode is over-utilized.

Different solutions are considered in the battery community to remedy this issue, one being to add a secondary coarse  pore network in top of the existing finer pore network of the electrode microstructure, the idea being lithium ions will diffuse easily through these highways oriented along the electrode thickness, following the path of least resistance, and then use a fine-pore network to reach the reaction site.

This electrode architecture is not only theoretical, as several groups have already demonstrated it experimentally. The selection of reference is listed here.

Just above you can see images obtained with scanning electron microscopy. These are cross sections of an electron, the vertical direction being the electrode thickness. And lithium ions are diffusing from top to bottom, or from bottom to top, depending on whether we are charging or discharging the cell.

On this image you can clearly see the coarse pore network, constituted of these large oriented channels here and here. And this electrode has been manufactured at the Lawrence Berkeley National Laboratory by Marca Doeff, with freeze tape casting. And you can find more details about the different existing techniques to manufacture such architecture in the article I spoke of earlier.

The ionic diffusion is model through a normalized through-plane diffusion coefficient, which is a ratio between the apparent or macroscopic diffusion coefficient of the porous electrode and the diffusion coefficient of the dense electrolyte. Therefore, this value is adimensional and we want to increase it.

It is related with the microstructure porosity epsilon and what is called tortuosity factor tau. Tau denotes the effect of the convoluted tortuous path of the pores that hinders the lithium ion diffusion. You want this parameter to be the lowest possible, ideally one, which is its theoretical lower bound. What is not apparent in this expression is that Tau depends on the porosity, usually through this relationship, which is a generalized Archie's relationship, an empirical or generalization of the Bruggeman analytical law. Other expressions exist to define this parameter, but it is out of the scope of this paper.

Two parameters are added:  alpha and gamma, that are specific to the type of electrode microstructure we are investigating. The important thing to understand here is a lower porosity increase Tau and a higher Tau decreases the diffusion coefficient.

So, what is a catch? If we want to improve the through-plane diffusion coefficient let's build wide channels, right? Well, that's not that simple. When we add these large channels, we create additional voids that the electrolyte will fill. That means less volume and mass for the active material particles which are actually storing lithium, and then the energy you want to put in your electric car. Then if you want to keep identical the total mass of the active material between the reference electrode on the right and the electrode with the secondary pore network in the left you have two solutions.

The first consists in increasing the thickness of the electrode with a secondary-pore network. The problem with this option is while you will keep identical the theoretical capacity, the gravimetric specific theoretical capacity, that is Ampere hour per kilogram is decreasing, since electrolyte weight is increasing for the same mass of active material. Similarly, the volumetric specific theoretical capacity that is Ampere hour per cubic meter is decreasing, since electrode volume is higher for the same mass of active material. Depending on the context, that might be fine if your battery system is not constrained by volume or mass. A more meaningful comparison is to compare with the same gravimetric specific theoretical capacity and the same volumetric-specific theoretical capacity. This can be achieved by keeping constant the electrode thickness and densifying, which means reducing the porosity, of the microstructure matrix, so the primary region in our model, to accommodate for the loss of actual material induced by the introduction of these coarse channels.

Well, then let's do this:  what is the problem? We can densify electrodes by calendering them, for instance. But do you remember when I said that decreasing the porosity is increasing the tortuosity factor which results in a lower diffusion? This is the catch. By adding these coarse channels we are improving the through-plane diffusion of the whole electrode, but we are degrading the in-plane diffusion within the microstructure matrix, since the porosity in the primary region is lower.

That's an issue because if the lithium ions can't reach the center of the primary region we'll have, again, a problem of partial utilization of the active material. This model is all about finding the correct dimension in order to balance the through-plane diffusion improved by the secondary pore network and the in-plane diffusion degraded by the secondary pore network.

Concerning the model itself it is quite simple:  there is no electrode chemistry, as we focus only on transport properties. Then this model must be considered as a pre-screening tool, you can use to quickly investigate very large design and parameter space as it is very fast, and then investigate some promising set of parameters that you have identified with this model, with a more complex numerical electrochemical model, which is slower.

This being said, the main elements of the model are listed here, although I will not detail then to keep the demonstration short. Of course, if you are interested about the details you can read this section or quote the article.

What is more important to understand is what are we doing with this model. How do we interpretate the model results to choose an optimal design? For this we have to define optimization objectives. Here three are considered and are listed here.

The first one consists in answer of this question. For a given transport coefficients and porosity what is optimal design of the secondary pore network that will maximize the through-plane diffusion gain and minimize the in-plane diffusion loss? The second optimization objective answers another question:  What parameter enforce the diffusion coefficient to be isotropic? The last one wants to enforce the isotropy of the characteristic diffusion time, thus adding to the equations, the electrode thickness.

Here I should highlight that the through-plane diffusion is a apparent through-plane diffusion, that is the through-plane diffusion of this whole electrode, while the in-plane diffusion is in-plane diffusion within the primary region, that is, from the interface between the fine and coarse network, to the center of the fine-porous microstructure.

These three different optimization objectives are not equivalent but focus on the same idea:  try to balance the improvement of the through-plane diffusion with the degradation of the in-plane diffusion.

Then, how to use in practical the model. First, we have to provide some parameters. Because the model is fast, being analytical, we can ask the model to run multiple times within a given parameter range to draw nice-looking result maps.

Here I will set up a test case for which I want to investigate total porosity between 0.3 to 0.4 with a grid resolution of 200. That means the model will run with 200 different porosity values, ranging from this minimum to this maximum. Here we neglect the impact of any additives, so I keep this value to be zero.

Then I have to set up the tortuosity coefficients. Here we are using slightly more complex tortuous porosity relationship, as we have this additional parameter, and these are linked to the impact of the additives, and because I choose to not model additives for this simple example I can ignore these two values. Instead I have to provide the value of gamma, here, and alpha, here.

For this example, I will only specify a unique value; you can see that my minimum is equal to my maximum, and because of this I can ignore the value I put on the grid resolution. I can also specify if the microstructure is intrinsically anisotropic. By this I mean that this, within the primary region we may have some electrode microstructure that have different transport properties along the in-plane and along the through-plane, which is typical for electrode battery microstructure.

Here I specify an anisotropy of 1.5, that means a through-plane tortuosity factor is 1.5 times higher than the in-plane tortuosity factor. That means the in-plane diffusion is better than the through-plane diffusion.

Next is the choice of the secondary pore network. You can either choose rectangular channels, so if you're on the left, or cylindrical channels, the figure on the right. We can choose to specify the dimension using either what we call the secondary region volume fraction, Rv; Rv is the fraction of the volume of the secondary region over the volume the primary region and the secondary region, so basically the volume fraction of the secondary region over the whole volume of the electrode.

Alternatively, you can choose to set your parameter using Rw, which is the secondary region width to primary region width – it is omega 2 over omega 1. So let's see these two parameters. Omega 2 is the width of the secondary region; omega 1 is the width of the primary region. So you can choose between a rectangular or cylindrical channel by clicking on this drop down button. For this example, I will keep the rectangular case.

The value here shows Rv from 0 to 0.25. So, value of zero means no secondary pore networks; that's our reference. And I will ask a grid resolution of 500. So I didn't mention it but you can change the value, either by moving the slider or by enter directly as a value in the edit box field.

Lastly, you can control the ratio between the width of the primary region over the electrode thickness, capital L. So again, omega 1 is this width and capital L is the electrode thickness.

Here I choose a lower bound of 0.01, that means the thickness is one order of times larger than the porous matrix width, and an upper bound of 1, so the two distances are identical, and I choose a resolution of 200. So, we have a resolution of 200 for the porosity, 500 for the secondary pore network volume fraction, and 200 for the width to thickness ratio. That means the model will run 20 million times and it will be done on my laptop, and it will use less than one minute because the model is analytical and then super-fast.

I can also choose to investigate a uni-layer case, that's as the figures that I showed before, or a bi-layer architecture, which his shown here. This basically is an alternative design of this secondary pore network architecture. If you choose a bi-layer case the model can only bound the actual result for reasons explained in the article. So for this example I will keep it simple and I will keep the uni-layer case, which can be done by keeping this ratio equal to one.

Lastly, I have to choose my method of comparison between the structured electrode with secondary pore network and the reference electrode, so I can compare the same thickness, so in this case the porosity is reduced within the microstructure matrix and the comparison is done at the same volumetric and same gravimetric theoretical specific capacity, or I can choose to keep constant the porosity between the reference electrode and the porosity of the primary region. In this case the thickness has to be increased but both volumetric and gravimetric theoretical specific capacity will be decreased. For this demonstration I will choose a more meaningful comparison case as the same thickness case.

Lastly, there is a special case where you can specify that the primary region are fully dense. This is an extreme case, but that can happen if the electrode undergoes a post- processing sintering step. And in this case the model is actually simplified and you only have to provide some dimension, width and thickness, plus the solid state diffusion coefficient and the electrolyte diffusion coefficient. I will not consider this particular case in the demonstration, so I don't check this box.

So, once you are here you can run the model by clicking the run model button and once it's done you can click the plot figures button to see the results. Or alternatively you can simply click on the green arrow run button, which I'm doing.

While the code is running, I have some time to show you figures options. Here you can choose is you want to display the secondary pore network design recommendations using volume ratio, Rv, or the width ratio. So if you choose Rv – and that's the option what is chosen by default, then the choice of the rectangular or cylindrical pore channels doesn't matter. However, if you select the width ratio then the channel geometry will modify a lot the design recommendations. And this is explained in detail in the article.

So, you can see the model is done; it has used 10 seconds. And now what is remaining is to the CPU time to plot the figures. I'll skip this option and I just want to mention that here you can choose some cosmetic option for how you want to display your figures, how do you want to save your figures, and where the figures will be saved.

So, our results are plotted below. First is the volume fractions within the primary region. So here you have the porosity within the microstructure matrix of the primary region. And as you can see, if we increase the volume of secondary pore network, that's the Y axis, we have to densify the microstructures – the porosity is decreasing. Here you have the primary region volume fraction for the active material, for the CBD, and this one is zero because we choose not to have any additives. And these are the volume fractions for the reference electrode.

Then we have the electrode thickness. Here, because we have chosen to compare the same thickness, the thickness ratio between the structure electrode and this reference electrode is one, so there is nothing to discuss. Below, you have the transport properties of the reference electrode. That gives you an idea about what are the actual tortuosity and diffusion coefficient rather than rely only on the coefficients, and on the title you have the expression used to calculate tortuosity. You can note here that the in-plane tortuosity factor is lower than the through-plane tortuosity factor, and that's because we have chosen this anisotropic ratio.

Below you have a dimensional – you have two-dimensional map, showing the transport properties of the structured electrode, so that with – the electrode with the secondary pore network, and this is as a function of the reference electrode porosity, or the total porosity, if you prefer, and the secondary pore network volume ratio, as I choose to use this metric to show my result and not the width ratio.

So within the primary region, here and here – this is for the in-plane; this is for the through-plane diffusion you can see that the diffusion is getting worse when we increase secondary pore network ratio, while when we look at the overall through-plane diffusion coefficient of the structured electrode we have the opposite trend:  here it is getting better. So that's the tradeoff we have discussed earlier between the degradation of the in-plane and the improvement of the through-plane diffusion. Below you have the same information except this time we plot the tortuosity factor rather than the diffusion coefficient, so we have exactly the opposite trend.

So, I spoke about different optimization function before, and here you have the result for the first optimization function. We try to maximize the in-plane diffusion gain on the left, the through-plane gain in the middle, and the result is the product of these two maps; we try to have the maximum.

The solid black line is the unique value, so everything above is below one, so you want to avoid this region because you have overall negative gain for your diffusion, and below this thick line is value above one, so you have an overall positive gain for the diffusion, so you want to be in this region. And the red line is the optimal region, so for let's say you have a porosity of 35 percent for the reference electrode; if you choose a secondary pore network volume fraction ratio of something like close to 0.9 then you will have an overall gain of around 30 percent for your diffusion.

Below you have the design recommendation for two other optimization functions. On the left what we want to achieve is isotropy between the in-plane and through-plane diffusion coefficient. Well, that does not depend on the width to ratio parameter. On the right we want to achieve isotropy between the in-plane and through-plane characteristic diffusion time, and this one depends on the width to thickness ratio. Also you can note this here, we have some empty space, and this is because when I chose my parameter range I've chosen the upper bound for Rv of 0.25, so the 0.25 is this line. So here it doesn't mean there is not analytical solution, there is an analytical solution, just if you want to see this region you will have to rerun the model with a higher upper bound for Rv.

Then the question is how to choose between the different optimization recommendation, so there is not a unique response to this question, but what I recommend is to consider the last optimization objective, this one, as it constitutes the largest set of variables for its calculation, especially the width to thickness ratios, it considers the thickness, and the tortuosity anisotropy, while these two are insensitive with – I mean this one is insensitive with anisotropy ratio and the thickness, and this one is insensitive with the thickness ratio.

For instance, let's say I consider an intermediate porosity of 0.3 and a width-to-thickness ratio of 0.5. Then the recommended value is around 0.2. For this value the characteristic diffusion time is identical between the through-plane, considering the whole electrode, and the in-plane, considering only the porous matrix. Although it does not tell me how much I've improved or degraded my overall diffusion. For this I have to report these values on the previous optimization objective result.

The model predicts that the in-plane diffusion will be significantly degraded, so at 0.3 and 0.2 we have – we have 0.18 times lower, while the through-plane diffusion is significantly improved around higher than four times, and the overall gain is around 0.65. Also, if I want to achieve the best possible diffusion trade-off I will ignore this recommendation and I will only choose a point along this red line.

This is only one example of the different parameter analysis you can perform with this model, and for instance, you could have set – let's say you could have set a constant porosity, but a variable transport coefficient and variable anisotropic coefficients.

If you want to see more results or a better understanding of this model or its background I recommend you read the articles mentioned in the beginning of this video, this one, and with this feel free to contact me if you have some suggestions or comments using this email address. So, thank you for your attention and have a good day.