Transcript
DOUG COBOS 0:10
Okay, so here’s what I want to talk about today. First, I want to very briefly cover the basics of soil water characteristic curves. If you guys are here, then you probably already know — oh, sorry, a couple of technical difficulties — some of the basics of the soil water characteristic curve, but I’m pretty sure that you guys all know about that. But then I want to focus in mainly on the dry end of the soil water characteristic curve, and then talk for a little bit about what you can do with the information contained in that dry end. And then I want to zero in a little bit and talk about really the meat of the presentation, which is the concept of a single point soil water characteristic curve. And then I want to talk for a little bit about a study that we did trying to validate that concept of the single point water characteristic curve, and then leave you with a couple of take home messages.
DOUG COBOS 1:03
So let’s talk in general about the soil water characteristic curve, or the moisture release curve or the water retention function, the pF curve, moisture sorption isotherm. I actually will talk about it in terms of soil moisture characteristic curve. That’s one of my favorites. But those are all names for the same basic relationship. And that is the relationship that you can see in the graph there between the water content of the soil and the water potential of the soil. Water potential also has a lot of analogs — water activity, geotechnical engineers typically talk about it as soil suction, pF, chi is the thermodynamic term, but those are all terms that relate to the energetic status of the water. And so what you’ll notice in the graph is that this has three generic moisture characteristic curves, one for a sand, one for loam, and one for a clay. And you’ll notice that those are all different. And each individual soil has its own individual soil water characteristic curve or moisture characteristic curve. And if you can generate that curve, then that gives you a lot of interesting information about the soil that has some really useful applications. So generating that moisture characteristic curve, half of that’s really easy. So measuring the water content of a soil sample or a bunch of soil samples is quite easy. You weigh a soil while it’s moist, you dry it out, and then you weigh it again, and that gives you the water content, either on a gravimetric basis or volumetric basis, doesn’t really matter, in this case. It’s the measuring of the water potential or the soil suction that is difficult. There is really no single instrument that can make accurate measurements of soil suction or water potential all the way from the wet end to the dry end. Typically, the wet end measurements have been relatively easy, but the dry end measurements have been pretty difficult to make. Just for an example, Gaylon Campbell and Sho Shiozawa published a paper in 1992 that had moisture characteristic curves in the dry end for a few soils. And there was so much pent up demand for those data that that paper has been cited, well over 70 times. And a lot of other authors needed those data and borrowed those data and have used them and so you’ll see those moisture characteristic curves in all kinds of different papers out in the soil science literature. With the introduction of the WP4, WP4T, and now the WP4C dewpoint instruments, those dry end soil water characteristic curve data became a lot more accessible. And so there was a proliferation of the information on the dry end of the moisture characteristic curve. And now we’ve taken the next step and developed the Aquasorp and VSA or Vapor Sorption Analyzer instruments that are automated instruments that give you that dry end of the soil water characteristic curve, and so you’re seeing another quantum jump in the information that the geotechnical engineers and the soil scientists are pulling from the dry end of the moisture characteristic curve.
DOUG COBOS 4:26
So let’s get back to a typical soil water characteristic curve. And I want to point out just a few things. First, I want you to note that this is plotted with a linear axis on the water content axis and a logarithmic scale on the water potential axis, so you’ll see 0.1, 1, 10, 100, 1000. This is a typical way, a typical format to plot these curves in. You’ll notice that at very high water content and very high water potential, there’s a zone of saturation even with decreasing water potential, this is until you reach the air entry point. Once you reach the air entry potential, then you have a large loss in water content as the capillary water drains from the soil pores. Once you get to the dry end, okay, these wet end data were generated with the UMS HYPROP and the dry end data with WP4C dewpoint instrument. Once you get to the dry end, you get a zone where there’s very little change in water content with large changes in water potential. And this is because the water is sorbed to individual particles and no longer contained in the capillaries.
DOUG COBOS 5:39
So what we want to zero in on today is this dry end of the water characteristic curve, where we are not talking about capillary water, but we’re talking about water that’s associated with the individual soil particles. So if we zero in there and continue to use this semi log plot with the linear scale on the water content axis and the logarithmic scale on the water potential axis, what do you notice? Well, you notice that this moisture characteristic curve becomes a straight line. And that’s pretty handy. Well, this has happened with all soils. Well, it pretty much does. Here’s a fine sandy loam. Here’s the silt loam soil. And so you’ll notice that this straight line behavior is pretty much universal amongst soils. This is pretty handy, because if you have this linear plot, you can easily fit that and calculate the slope of the semi-log moisture characteristic curve. And the slope of that curve contains a lot of information that I want to talk about a little bit now.
DOUG COBOS 6:44
Let’s talk about some of the things you can do with the dry end of that moisture characteristic curve in general, and also with the slope of that semi-log plot. So the first thing that you can really mine from those data is that you can determine the soil specific surface area, just from the dry end of that soil moisture, soil water characteristic curve. It’s highly correlated to the slope of that semi log plot. So back in 2006, paper was published that gives an extremely simple relationship between specific surface area, where the specific surface area is just a scaling factor of 1.84 multiplied by the mono layer coverage of water on the soil particles. And then that’s, of course, multiplied by the slope of that semi log plot in a format that is relevant thermodynamically. So I’d encourage you to look at that paper if you’re interested in that. In 2011, another group of soil physicists did a similar study and came up with a slightly different relationship because the formats of those of the moisture characteristic curve are a little bit different, but it is pretty clear that the dry end of that soil water characteristic curve will give you a lot of information about the specific surface area of your soil.
DOUG COBOS 8:06
Many of us are also familiar with expansive soils or shrink swell soils, especially those of us who are into geotechnical engineering and soil mechanics and foundations and things like that. It is pretty important when you’re building a foundation or a road on a soil that you know if that soil is an expansive soil. And so Gordon McKeen in 1992, notice that the coefficient of linear extensibility, which tells you about the expansive potential of a soil, well, he noticed that that coefficient of linear extensibility is highly correlated to, again, the slope of that semi log soil moisture characteristic curve. And so he developed a framework that’s used pretty often by the geotechnical engineers to determine if a soil is an expansive soil or if it’s not. And let me show you that just real quick. If you plot the soil water characteristic curve in this format with water content on the x-axis and then the log of water potential, or the log of suction, on the y-axis, pF is simply the log of water potential, so this is that semi log plot that we’ve been talking about. Then of course, these become a straight line. And this soil for instance, this is an L soil which is a very sandy soil, this has a slope of negative 97, okay, which is in this less than negative 20 case. And so that’s a non expansive soil. And I don’t think any of us would expect a sand to be an expansive soil. As you go to the Palouse, this is a Palouse silt loam, you’ll see that the slope is much higher but still in that non expansive category, which is what we’d expect here. But then as you go to the B horizon and get down into some of the clay layers in this Palouse soil, this has a slope of negative 17. And so we’re getting into the low expansiveness category. And if you were to start dealing with some clays from Texas or Mississippi or Colorado, some of the 2:1 smectitic clays that are expansive, then you would see a plot that is up here with a less negative slope to it. So this is a pretty successful method of determining the expansiveness of soil without having to do other geotechnical type testing on it.
DOUG COBOS 10:30
Other folks have used the dry end of the soil water characteristic curve to look at gas movement, gas diffusion in soil, both in terms of pesticide volatilisation and remediation of VOCs. And there’s also a growing body of knowledge that says, that at least indicates that you can predict and understand the cation exchange capacity of soils by using information that’s contained in the dry end of the soil water characteristic curve. And this is an area that’s getting some active research right now, both in the soil physics area and in the geotechnical engineering area. So the take home here is that there’s a lot of information and a lot of different things that you can do with that dry end of the moisture characteristic curve.
DOUG COBOS 11:17
So now I want to get back to the shape of that curve and talk a little bit about the single point soil water characteristic curve, which is what I was hoping primarily to talk about today. Okay, so here are three soils, couple of sandy soils and and a silt loam soil. And you can see that we’ve again plotted these on that semi log plot with water content versus log of water potential, and these become a straight line. And we all remember that, you know, just from our introductory math classes back in probably junior high that any two points define a line. So I will tell you that myself, I have spent quite a bit of time generating these moisture characteristic curves, and I always collect a bunch of data. I mean, these are data that I’ve collected for various reasons, where I generally have 5, 10, 15 points in the dry end. Well, what I’m telling you is that you don’t necessarily need that many data, okay. If these are always linear plots and two points define a line, then all you really need is maybe a point at air dry, okay, and then another point wetter in the dry end still of the moisture characteristic curve, and those two points define the line and you don’t need all the rest of those points. And so you could do this at any two points along the line. So we’re already establishing that you don’t need to collect a whole bunch of data — really, all you need to collect are two points. But I keep talking about a single point moisture characteristic curve. So let’s look at that.
DOUG COBOS 12:49
What else do you notice about the two trend lines that are fit through these data? Well, if you look closely, you’ll notice that if you extend those and you extrapolate all the way to zero water content, then all of these extrapolate through the same point, okay, negative 1000 megapascals of water potential or positive 1000 megapascals of suction at zero water content. Now, I want to point out that you can never reach this point. You can never actually reach zero water content on Earth. There’s always going to be some water left even after you bake the soil out. But this is an extrapolation, okay, an extrapolation to the point where we would reach zero water content, and that is always negative 1000 megapascals, negative a million kilopascals. If you’re used to log kPa that’s 6.0 log kPa, seven pF, negative two for a chi value, that’s a thermodynamic value that is actually the log of water potential, or 0.06% relative humidity. Those are all the same value of negative 1000 megapascals. So now, if we have our point pegged for all moisture characteristic curves at zero water content and negative 1000 megapascals, then if two points define a line, and we already have one of those points, then it’s pretty easy to grab one point on that moisture characteristic curve and reproduce the line that defines that soil water characteristic curve. And you could do that again with just an air dry point that’s at about negative 100 megapascals, but then you’d be extrapolating quite a bit out here. I would tend to do this at a little bit wetter point, if possible, maybe wet point out here. And then you’ve again defined your line and determined the entire dry end soil water characteristic curve with one single measurement, okay, one single water content—water potential pair. Pretty cool, huh? Well, we certainly thought so and still think so to some extent. But there may be some issues with that, that we’ll get to a little bit later.
DOUG COBOS 15:14
Wanted to point out that this is not necessarily a new technique. Bill Likos back in 2008 published a paper on the water content 75 technique. And so, what is this? Well, what his paper was saying is that if you equilibrate a soil sample over saturated sodium chloride, which has a relative humidity of 75%, or a water potential of negative 40 megapascals and then you measure the water content when that soil has been equilibrated at negative 40 megapascals, then you can use that to classify an expansive soil. And I would say that this is simply in an extension of the McKeen technique and an example of the single point moisture characteristic curve. So you can see I have some water content for various soils at this 75% relative humidity or negative 40 megapascals plotted, okay. If you use that single point, and recreate the single point moisture characteristic curve by going out and using the second point at negative 1000 and zero water content, then you’ve recreated these lines, and taking the slope of these lines will give you the expansive potential by using a technique like that McKeen technique. So this is essentially a one point soil water characteristic curve, and just taking the slope of that and plugging it into the McKeen framework. So it’s not a new concept. It’s been out there, but maybe not pointed to explicitly.
DOUG COBOS 17:00
So we wanted to look at this, you know, we’re kind of intrigued by this. And so we wanted to look at this in a little bit more detail using the new tools that we have available to us. And so we did a study in preparation for this last Soil Science Society of America meetings in Long Beach. I guess it was last month in fact. And so what we did was we grabbed those four soils from that well known Campbell and Shiozawa paper and we also got 13 well characterized soils from the Texas A&M University soil library, and we also included bentonite in the study. And the objectives were first of all to take a closer look at this zero water content intercept concept that that we had been observing for many years, and then also to revisit this expansive soil characterization framework of McKeen. And why would we care to revisit this McKeen analysis? Well, there are some things that we saw in that paper that didn’t really make a lot of sense. McKeen’s original paper points to a zero water content intercept of 6.25 pF, which is negative 174 megapascals. If you’ll remember, we’ve been identifying that zero water content intercept value as negative 1000 megapascals, which is nothing like negative 174 megapascals. In terms of pF, McKeen’s pointing to 6.25, this value of seven right here would be the negative 1000 megapascals. And this of course, is a pretty big difference. And the reason that we think that there is a difference there is because McKeen’s original work used the filter paper technique to measure soil suction or water potential to generate those soil water characteristic curves. And this is a very indirect method that has all kinds of problems associated with it. And so it’s not surprising that the values that were found there were different.
DOUG COBOS 19:03
So what we wanted to do for this study was use a new tool that’s available to us now. This is the vapor sorption analyzer, which will create the soil water characteristic curve in the dry range, fully automatically. Okay, so you put a soil sample in and it varies the water potential, automatically and measures water content and water potential while it’s doing that, and so it generates a very nice soil water characteristic curve. Another advantage is that it generates both the wetting and drying legs of the hysteresis loop. So you get the entire hysteresis loop from this instrument. And it gives you a lot better data density than were previously available. And what do I mean by data density? Well, the curves that I’ve been showing you, or those straight line plots that I’ve been showing you, I’m saying you can do them with one point. Those curves contain something like 10 points, okay. This gives you hundreds of points. And running some soils through this and running some of these soil water characteristic curves with better data density, it became apparent that there are things going on there that we weren’t able to see before with the older techniques. So we wanted to do a little bit better job of looking at what those are.
DOUG COBOS 20:22
So here are some results. And here are some data from the VSA. And you’ll see that we’ve got some curves there. The dots are real soil data from the VSA, maybe 100 points or more on one of the curves. And these happen to be the adsorption legs of those curves. And if you squint a little bit, you can draw straight lines through those curves, and you can extrapolate those out to that value of zero water content and negative 1000 megapascals. So, before we get into why you have to squint, wanted to point out that this McKeen value of negative 174 megapascals for the zero water content intercept is certainly not right, okay. That — I can’t remember the number. I think that relates to something like 24% relative humidity. okay. And it’s well up the scale of water potentials that we might expect. And so certainly, there were some big errors in his original analysis — I think based on the fact that he was using filter paper instead of maybe a vapor pressure or dew point technique for that.
DOUG COBOS 21:35
So we’ve already determined that that 6.25 pF or negative 174 megapascal value is wrong. But let’s take a closer look at some of these curves that we’re using here and see what jumps out at us. First of all, you’ll notice that there are hysteresis in these curves. So if you just look at the green curve, you’ll notice that there’s a relatively large difference between the adsorption curve, okay. If you take any water potential, okay, and look at the adsorption curve, then you have a water content here, okay. If you look at the desorption curve, then you have a significantly higher water content. And this hysteresis between the adsorption and desorption is something that is pretty much ubiquitous in natural systems. You always see hysteresis and especially, you will see hysteresis in moisture characteristic curves, if there’s a lot of clay in there. One thing that was interesting is that in the soil science literature, you will see references that say there’s no hysteresis in the soil water characteristic curve of a sand. And so here’s a sandy sample. And I would say that, of course, if you blow this up, you will still see hysteresis even in a sand. And this is pretty interesting, because the mechanisms for this aren’t well documented. In a clay you’ll get hysteresis from the differences in adding water to the inner layer, interclay layer spaces, versus the energy required to remove that water from those spaces. But in a sand sample, you don’t have these clay layers to deal with. And this is just water that’s absorbed on the exterior of some mineral particles. And so there’s not really a great mechanism for hysteresis there, but there is some indication that this could be created by the cation exchange sites, which are few and far between on sand particles but are still present. You certainly would see hysteresis in the wet end of the soil water characteristic curve due to the differences between filling and emptying pores and ink bottle effects and things like that. But that’s not what we’re talking about here. What we’re talking about here is water that’s associated with the surfaces of these soil particles and not not capillary water. So this is actually a pretty interesting result to notice that we get hysteresis even in sand samples.
DOUG COBOS 24:07
So what are the implications of this hysteresis? Well, if you look at these moisture characteristic curves that I have plotted, you will note that there is a fundamentally different soil water characteristic curve for the soil if you’re adsorbing water versus if you’re desorbing water. And this is something that hasn’t really been paid much attention to. In the soil physics literature or in the geotechnical engineering literature, people will say, yes, I created a moisture characteristic curve, and here are the characteristics of it, here’s the slope. But you really need to specify if that’s an adsorption curve or a desorption curve. In terms of the McKeen framework for expansive soils, if you’re looking in this soil, for instance, at the adsorption curve, you get a slope of negative 10, which puts this in a medium expansive category. If you’re looking at the desorption curve, you get a different slope, You get something that’s less than negative nine, and that puts it into the high range, okay. You’ll notice that in the sandier sample, you jump from a non expansive on the adsorption loop to a low expansiveness on the desorption loop. So take home message here is that you need to pay attention to and you need to specify if you’re dealing with the adsorption leg or the desorption leg of that hysteresis loop because you could come to different conclusions based simply on the history of the sample, whether it’s getting wetter or whether it’s getting drier during your your measurement.
DOUG COBOS 25:45
The other thing that we noticed and this is really why you had to kind of squint at those curves that I saw to be able to fit them with the straight line is that there are inflections in the dry end of the soil water characteristic curve, especially in the high clay samples. You’ll note that they’re inflections, these all occur, in this case, at about the same water potential, and these are probably due to water being adsorbed into the interlayer spaces. As the clay expands and water goes between the clay layers, then then you’ll see a difference in the energy state necessary to get the water in and out. And so you see these inflections. What are the implication of these inflections? Well, if you’re using that single point soil water characteristic curve method that I was advocating earlier, and you collect your one point here at something like air dry, you’re going to get a different soil water characteristic curve than if you grab your single point over here. You’ll notice that these lines are quite different from each other, okay, and have different slopes. So you’re quite likely to have a different conclusion when you’re using your specific surface area analysis or your expansive soil analysis. So you have to be careful when you’re working with high clay soils, and especially with your 2:1 clays that are shrinking and swelling. And interestingly, this inflection hasn’t been observed before the VSA instrument. If you’re doing this with a WP4C, then you have only maybe 5 or 10 points along this curve, and it’s real easy to miss these inflections. So this is something that the VSA instrument has allowed us to look at in a lot more detail. And there are groups out there that are looking at these relationships and really putting mechanistic effects to explain what’s going on here.
DOUG COBOS 27:47
Now let’s look at the most extreme example. And here’s a bentonite. You’ll see both the wetting and drying legs of the hysteresis loop on this bentonite, which is an extremely swelling, 2:1 clay. And so what do you notice here? Well, if you, depending on if you’re looking at the adsorption loop or the desorption loop, and depending on if you’re looking in kind of the wetter portion of the dry end, or the very driest portion of the dry end, you can get a zero water content intercept of just about any value that you want, okay. If you plot this portion, you end up down here. If you plot this portion of the desorption loop, you end up with a different number. If you plot a little bit wetter portion of the desorption loop, look, you end up out here at maybe, you know negative 10,000 megapascals instead of our zero water content intercept value of negative 1000. Interestingly, if you’re on the adsorption leg in the driest portion, this does extrapolate out to our zero water content intercept value, but that’s not something that I would hold, you know, hang my hat on. So this kind of underscores the difficulty with the single point water characteristic curve.
DOUG COBOS 29:05
And so let’s sum up what we talked about today and try and give you some important take home points. The question that we wanted to talk about today and the question we wanted to answer is, is the single point soil water characteristic curve concept valid? And the answer there is, sometimes. If we had been talking about this, you know, four or five months ago, before we collected these data with the VSA, I would have told you, yes, always, because every soil I had looked at had that zero water content intercept of negative 1000 megapascals. But in light of this new information, with better data density and some new techniques, then I would back off that and I would say that it’s probably useful in low clay soils in the dry end, but be careful when you’re working in high clay soils and especially when you’re working in 2:1 clays. One thing that is clear is that McKeen’s value of negative 174 megapascals is wrong. That 6.25 pF value for the benchmark zero water content intercept is not correct, and that value is generally much closer to negative 1000 megapascals.
DOUG COBOS 30:21
One other take home point is that hysteresis produces fundamentally different soil water characteristic curves on the adsorption leg versus the desorption leg in all soils, not just in high clay soils, but also in sandy soils. And this is something that needs to be — attention needs to be paid to this. And we need to specify this when we’re talking about any analysis that involves the soil water characteristic curve. And then finally, I wanted to point out that there’s a lot of non linearity and inflections in the soil water characteristic curves in the high clay soils, and especially in the 2:1 clays. And this really confounds the single point soil water characteristic curve method, something that we hadn’t seen before, but certainly something that is something that we need to keep in mind. So those are the things I wanted to talk about today. Hope you learned something and hope you found it interesting. I’ll be here for just a little bit to take some questions if I see some come through. And anything that we don’t get to in real time, I’ll email you afterwards with some answers to those. So thanks for attending.
DOUG COBOS 31:47
Okay, I have a question here. How long does it take the VSA to measure the soil water characteristic curve? That’s a good question. It depends a little bit on the soil type and how you configure the instrument. But typically, the curves like you see on the graph over here, 24 to 48 hours to generate the full hysteresis loop with the data density that you see there.
DOUG COBOS 32:22
Okay, I’ve got another question here. What’s the cutoff between capillary water and the dry end that you talk about? That’s a good question. So if you remember the one graph that I showed you that had the HYPROP data on the wet end and the WP4C data at the dry end, you’ll note that there’s a huge difference in the shape of those, and if we’re trying to look at the linear portion in the dry end, then to be safe, I think you typically want to be drier than about negative one megapascal. And I typically would go even a little bit drier than that to make sure that you’re in the dry end and out of the capillary water zone. A lot of times people will point to the end of the capillary water being closer to one bar or 100 kilopascals or 0.1 megapascals, but certainly negative one and drier will get you out of the capillary water zone.
DOUG COBOS 33:28
There’s another question asking, why have we not seen the hysteresis effects on the dry end before? And that’s another good question. And I think the answer to that is that we just haven’t collected hysteresis loops. So with the nonautomated methods, it’s a lot more difficult to collect data on the desorption portion of the curve. Typically when I generate a soil moisture characteristic curve with the WP4C, I start with you know air dry soil and add water, different amounts of water and create a set of samples along the moisture gradient. And so I start with dry soil and add water, so I’m on the adsorption leg, and it’s quite a bit more difficult to create samples along a moisture gradient on the desorption leg. We’ve done it. Start with a wet sample, put it in front of a fan, let your samples dry for different lengths of time, and then recap them, and let them reequilibrate. But it’s a more laborious process, and it’s a bigger pain in the rear to do that, so typically that hasn’t been done. And nobody really looked at that in the past, but now with the ability to do that automatically in a relatively short amount of time, then those data become much more accessible. Let me spread this out a little bit.
DOUG COBOS 35:00
This is another good question. Okay, so someone’s asking, they’re saying that I thought the zero moisture intercept was fixed based on a thermodynamic calculation at oven dry temperature of basically 105 C, which is what we typically dry soil at. That is a really, really good question. The suction or water potential that you come to at oven dry is a fixed value, if you control the vapor pressure of the air. So if you know the temperature of the sample, okay, which is 105, typically for drying soil, and if you know the vapor pressure of the air, then you know the water potential that you bring it to. But that water potential that you bring it to, and that state you bring it to, is not at all zero water content. Because, I mean, you could continue to heat soil, you know, much, much hotter and still have water associated with those soil particles. You can never bring a sample to truly zero water content, excuse me, to zero water content if you’re on Earth where we have water vapor. So the true zero water content intercept that we’re talking about doesn’t actually exist. It’s a fake point because we can’t get there. But you can extrapolate out to that point using those linear relationships. So that’s a really good question. So the thermodynamic basis of zero water content is invalid but for a fitting point, then that works pretty well.
DOUG COBOS 36:43
One other question, how does the McKeen expansive frame work? I guess, yeah, the expansiveness, how does that compare to the plastic limit and liquid limit? Okay, so this is a question about the Atterberg limits. Geotechnical engineers use the Atterberg limits, the plastic limit and liquid limit, and the difference in water content between those two estimate if a soil is expansive or not. So if it’s a plastic clay, then that means that it’s probably expansive, and if it’s nonplastic clay then that means it’s probably not expansive. That’s a really good question. Both methods, the McKeen framework and the Atterberg limits are indirect measurements of trying to estimate whether or not a sample will be expansive. Really, the way to determine that is with some true expansiveness tests, maybe a coal test or maybe one of the other geotechnical tests for that. So I think that they— my understanding from people who have looked at that is that those do line up pretty well, that the McKeen analysis and the Atterberg limits do line up pretty well. But there is variability. Some samples will show to be a plastic clay and not show up as expansive in the McKeen analysis and vice versa. There is some research being done now with the VSA instrument and geotechnical engineering that aims to answer a lot of those questions. So we have some friends that are working on those questions and comparing the Atterberg limits to the shape of the moisture characteristic curve along with some of the other geotechnical tests to try and really get to the bottom of the mechanisms that caused these different things.
DOUG COBOS 38:39
Okay, I think that’s about all that we’ve got for today. I hope you guys enjoyed. Thanks for tuning in. And my understanding is that you will get an email about this, and it will get you to the archive of this virtual seminar so that if you wanted to go back and look at it or send it to a friend, you’d be welcome to do that. So thanks a lot for participating and see you later.
