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# Weighted Averages: Averaging Averages or Rates

In our series on averages, last week we introduced the idea of the weighted average (or weighted mean), where each item has a weight attached. The classic examples all involve grade averages in various ways. This time, we’ll look at how weighted averages arise when you need to average several averages together, something we touched on last time, but which arises often in very different settings. One thing we’ll see repeatedly is that the appropriate average depends on your needs.

## Can you average averages, or  not?

We’ll start with a 1997 question from a professional who had learned that, generally, you can’t average averages, but didn’t know when you can:

Algebra: Average of an Average

Hi!  This one is a real-life need.  I work for a consulting firm, and am having a hard time remembering my basic Algebra!

If the "Wireless" line of business hires 50 people in one month, and the "Multimedia" line of business hires 80 people in one month, what is the average number of people per month we are hiring?

At first, one thinks (50 + 80)/2 = 65 is the correct answer, but it's not because we're taking the average of an average, right?

Don't you have to let x = something and then do 1/50 + 1/80 and solve... or something?

Hi Ron,

65 is the correct answer if the question is: "What is the average number of people hired per line of work per month?"

Here, we are just averaging the two averages “per line”, so as long as we state clearly what it means, it is valid. The question is, is this average what you want to know, or is it something else entirely?

Assuming that the entire company is made up of only those two lines of business, and you're asking "What is the average number of people hired by the whole company per month?", the answer is obviously 50+80 = 130.

This seems to be what Ron wants, and it isn’t really an average at all, just a total!

So what about the idea that you can’t average averages? That’s really a different story:

The operations above are perfectly reasonable. Let me show you the kind of situation that I think you were worried about, and you'll see why it's different from the situations above:

I take 4 exams and have an average score of 80. Then I take 2 more exams and on those 2, my average is 100. What's my overall average for the course?

The wrong way to do it is (80+100)/2 = 90. This is wrong because the averages were of different-sized groups. To get the correct answer, I know that on the first 4 exams I got 320 total points because when I divide 320 by 4, I get 80.

Similarly, for the last two exams, I must have gotten 200 points total. So for the six exams, I got 200+320 = 520 points and 520/6 = 86.66666 = my real grade average.

We saw this scenario last week: If we just average the two averages, the result can only be described as the average of the averages, not the average for the course. The latter has to be weighted, because for the course, each exam counts the same, not each of these two sets of exams.

As we saw last week, a weighted average can often be understood by breaking it down:

For your problem, the averages you are averaging are for the same period, so it works out. To convince you that it's true, let's just look at a situation where the averages came from 10 months of data.

Then in the first line of business, 500 people must have been hired, since 500/10 = 50. Similarly, 800 were hired in the other, since 800/10 = 80. Altogether, 1300 people were hired in the ten months, or 1300/10 = 130 per month, company-wide.  Or if you're trying to get the average per line of work, 65 is right, since if each group had hired 65 people each month for 10 months, there would be 65*20 = 1300 total hires, so it works out.

Bottom line: always think about what you want, and what an average means, rather than use an average (or not!) unthinkingly.

## Using a weighted average

Averaging Averages

I was reading your response on averaging averages. This has come up in a meeting I go to regarding a report that I am responsible for every month. After reading your response, I want to make sure that I am doing this right.

I have 5 different departments that send me an average rating to 3 different questions from a feedback form (rated 1-5). For example, this is what they send me:

Question 1 - average 4 (this is the average of question 1 from all feedback forms this department received during a certain time period)

Question 2 - average 3 (same comment as above)
Question 3 - average 5 (same comment as above)

Total average- 4

After getting this information from all 5 different departments, I combine them in a total company report by taking everyone's average to question 1 and then average that to get an average for the whole program. (For this program the company average to question 1 is...) and so on for questions 2 and 3 and the total average.

If I am reading your response correctly to the question on averaging averages, this is okay to do if all information is contained within the same time period (which it is).

Please advise.

Doctor Douglas answered, starting with a basic assumption:

Hi Stacy,

It sounds as though you are trying to average a set of averages. As long as the data reflect the same measurement (or question) for each of the individual groups, then it is okay to proceed. That is, if question 1 is the same among all departments ("Please rate the chef at the last company BBQ on a scale of one to ten.") then it is a meaningful question to ask: what did the employees think of our last BBQ chef?

Of course, it would be meaningless to average question 1 if each department had a different question, or if the responses were on different scales. But how shall we calculate the average?

Now, when you take averages of averages, there is often a preferred way to do this operation. It is called "weighted averages" and reflects the fact that the number of observations may vary among the different groups. For example, let's say that the Marketing Department has 50 employees, and the Research Department has only 6. If the Marketing Department average was 8.5 ("BBQ was great!") and Research Department average was 3.1 ("heartburn!"), what is the correct average for these 2 departments? We might simply take the average of 8.5 and 3.1, i.e. 5.8.  But it seems unfair to let the Research Dept sway the whole vote, having only 6 employees. The weighted average accounts for the differing number of employees:

weighted avg. = (no. of M)(AVG-of-M) + (no. of R)(AVG-of-R)
-------------------------------------------
(total no. of M and R together)

= (50)*(8.5) + (6)*(3.1)
----------------------
(50 + 6)

=  7.92

This is the classic example of a weighted average, and, as we saw last week, it amounts to finding the total score for the whole company by multiplying each average by the number of people it represents, and dividing by the total number of people. Thus this average is what we would get if we just averaged all the individual scores.

Do you see how the final average is much closer now to the Marketing Department's evaluation? Another way I sometimes explain it is that simple direct averaging is like the way states are represented in the U.S. Senate (every state gets two votes), while weighted averaging is like the way states are represented in the U.S. House of Representatives (every state is represented in proportion to its population).

As we’ll be saying several more times, each of these is a valid average, but has a different meaning. The U.S. Congress intentionally has one part in which every state, even the smallest, is treated equally, and another in which the largest state has more power, and each person is treated equally.

Now, it's impossible for me to say which type of averaging is correct for your situation, but I think it's probably better to use weighted averaging when you have information about the number of observations in each of the groups.

I would agree.

## Averaging percentages

A slightly different situation was involved in this 1998 question:

Averaging Percentages

Hi,

I am having difficulties in explaining to several friends that you cannot take percentages by totaling them up and then averaging the total of the percentages. It does not equal the percentage of the total of the numbers.

Is there a rule or theory that can explain this better?

Hi, Dominic. I'm not entirely sure what kind of problem you are referring to. Certainly there are at least some situations where you can average percentages. For example, if there are 50 questions on an exam, and three students got 20%, 30%, and 40% of them right, then the average number of questions they got right is 30%, or 15 questions.

This is like our situations above where we had equal groups, so a straight average made sense.

I suspect what you are thinking of is cases where the percentages are taken from different totals, in which case weighted averaging is needed. For example, if I survey 20% of 50 people, and 80% of 500 other people, then I have not surveyed (20+80)/2 = 50% of the total population, but:

.20 * 50 + .80 * 500   10 + 400   410
-------------------- = -------- = --- = 74.5%
50 + 500           550      550

The problem is simply that the percentages in such a problem do not represent fractions of the same total, so they can't be added.

As with an average of averages, here we are reconstructing the individual numbers (10 and 400 out of 50 and 500), then finding the actual percentages of the whole. That’s what a weighted average is.

I closed by referring to some of the answers we saw last week.

## Average price per square foot

For a specific real-life example, consider this 2003 question:

Average of Ratios vs. Ratio of Averages

I write and maintain software for real estate agents, and we include a calculation called, "Average dollars per square foot."  We currently calculate this as the ratio of the average price divided by the average square footage of all the homes in the list.  It seems to me that it should be calculated as the average of the price-per-square-foot ratio of each house. Can you think of any reason why the ratio of the averages would be more useful than the average of the ratios?  Is there a technical name for this ratio of averages?

I answered again, pointing out as before that different calculations can both be meaningful, though they mean different things:

Hi, Laure.

Interesting question!

What you are currently calculating actually does make sense; you are just averaging over all the square feet of houses, rather than over all houses, and that may be just the right thing to do -- or it might not.  Here's what I mean:

Suppose that N houses are sold; the sum of all their prices is P, and the sum of all their areas is A.  (That is, if the individual prices are P1, P2, ..., Pn, and the individual areas are A1, A2, ..., An, then P is the sum of P1 through Pn, and A is the sum of A1 through An.)

Then the average price of a house is P/N, and the average area of a house is A/N; and you are calculating

P/N
--- = P/A
A/N

as the average price per square foot.  And that is exactly what it is: the total price of all those square feet, divided by the number of square feet.

In terms of the individual numbers, this is $$\frac{P_1+P_2+\dots+P_n}{A_1+A_2+\dots+A_N}$$ Imagine laying out tiles representing each square foot of each house, and dollar bills representing the price of each house. This average spreads all the money equally over all the tiles, to find an overall price per square root. That sounds perfectly reasonable.

What you envision is

P1/A1 + P2/A2 + ... + Pn/An
---------------------------
N

which would average the price per square foot of all the houses.  This puts the focus on the individual houses, rather than the individual square feet.  How would this be different?

Again, this is $$\frac{\frac{P_1}{A_1}+\frac{P_2}{A_2}+\dots+\frac{P_N}{A_N}}{N}$$ Imagine laying out tiles representing each square foot of each house, keeping each house separate, and dollar bills representing the price of each house, spread equally among the tiles of that house. Now we take one tile and its cost from each house, and average those, spreading the money equally among the selected tiles. What have we found this time?

The way to see the difference is to take an extreme example. (Don’t check whether the sizes or prices are realistic; they aren’t meant to be!)

Well, let's take a simple case with N = 2.  Suppose we have a big, well-built house of 10,000 square feet, and that it costs $2,000,000 ($200 per square foot), and a little house of 1,000 square feet that costs $20,000 ($20 per square foot). Then the total cost of the houses P is \$2,020,000, and the total area A is 11,000 square feet. The average price per square foot is

P/A = 2,020,000/11,000 = 183.6

(closer to the more expensive price) while the average of the two price-per-square-foot numbers is

average(Pn/An) = (200 + 20)/2 = 110

(which is considerably lower).  What pulled the first number up is the fact that the bigger house had the bigger price per square foot; since we counted each square foot equally, the numerous high-cost ones won.  The second calculation treats all 10,000 of the expensive square feet equally with the mere 1,000 square feet of the little house, so the little house pulled the average down.

Which do you think is the better calculation?

Both numbers are meaningful.  The first fully deserves the name you are giving it (though there is definitely some ambiguity in the English!); but the second may better reflect what the average homeowner (as opposed to the "average square foot of floor space") can expect.  Call it, perhaps, the "cost per square foot of the average house", where the number you are currently calculating is the "average cost of a square foot" or "cost of an average square foot".

Perhaps the builder would consider the first calculation more appropriate, since they put more effort into the bigger house, and every square foot they built matters to them; but the homeowner might care only about his own house — which neither calculation really reflects!

So, again, both numbers can reasonably be called "average cost per square foot"; which is more useful to you depends on how you want to use it.  Do you want a number that is pulled up by big fancy houses, or one that shows what the average house is worth?  Or would separate numbers for different categories of houses make more sense?  Perhaps you can gather data that shows how costs per square foot are distributed, and how each average reflects that.

Any kind of average is an attempt to reduce a lot of data to a single number, and will never give a full picture of a diverse population. Other numbers, or a graph of the distribution, are often more useful.

Laure chose to leave the calculation as it was.

## Average speed

Let’s look at one more 2003 question, which is about a ratio in disguise:

Weighted Average of the Velocities

A truck on a straight road starts from rest and accelerates at 2.0 m/s2 until it reaches a speed of 20 m/s. Then the truck travels for 20 s at constant speed until the brakes are applied, stopping the truck in a uniform manner in an additional 5.0 s.

(a) How long is the truck in motion?
(b) What is the average velocity of the truck for the motion described?

Hi, Garin.

The trick here is to get the average velocity for each of the three phases of the trip, then do a weighted average of the three velocities.

Average velocity for phase 2 is easy, 20 m/s. During phases 1 and 3, the velocity changes linearly. That means that we can just average the starting and ending velocities to get the velocity for the entire phase.

This sort of average becomes most meaningful when you apply calculus, but for uniform acceleration, it requires only algebra. Doctor Edwin chose to leave that part for Garin to handle (or to ask for more help, which he never did), and focused just on the idea of the average. It turns out that the acceleration phase takes 10 seconds, at an average speed of 10 m/s; the cruise phase takes 20 seconds at 20 m/s; and the deceleration phase takes 5 seconds at an average speed of 10 m/s.

Once you've got those, you'll do a weighted average with respect to the time spent in each phase:

(v_1 * t_1) + (v_2 * t_2) + (v_3 * t_3)
v_avg =  ---------------------------------------
t_total

The calculation looks like this: $$\frac{v_1\cdot t_1 + v_2\cdot t_2 + v_3\cdot t_3}{t_1+t_2+t_3} = \frac{10\cdot 10 + 20\cdot 20 + 10\cdot 5}{10+20+5} = \frac{100 + 400 + 50}{10+20+5} = \frac{550}{35} = 15.7\text{ m/s}$$

Just for fun, I'll point out that the same set of equations has another interpretation that works just as well. Average velocity is just distance over time, right? So if we figure out how far the truck went in each phase, add them all up, and divide by the total time, you'll also get the same answer:

d_1     +    d_2     +      d3
v_avg =  ---------------------------------------
t_total

But the middle terms in those two equations are the same:

d_2 = v_2 * t_2

and so are the first terms:

a(t_1)^2
d_1 = --------
2

v
a  =  -
t

v_1_end * (t_1)^2
d_1 = -----------------
2(t_1)

v_1_end * t_1
d_1 =   -------------
2

But since the starting velocity is zero, we can add it in wherever we want:

(v_1_start + v_1_end)
d_1 =  ---------------------- * t_1
2

which means that our first terms are the same as well, and so are our third terms.

Anyway, there you have it. Total distance traveled divided by time, or a weighted average of velocities with respect to time, it works out to the same thing.

In other words, each “average velocity · time” is a distance, so our average velocity calculation was really just total distance over total time. We’ve seen previously that when distances are the same, we can use a harmonic mean to find the average speed; this confirms that when times are known but different, we can use a weighted arithmetic mean, using times as weights.

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Number of isoceles triangles

Three vertices are chosen randomly from the seven vertices of a regular 7-sided polygon. The probability that hey form the vertices of an isosceles triangle is:

I know that the this problem deals with geometry and combinations. I just drew its picture and from it I think that from a single vertex of the polygon 5 isosceles triangles can be drawn. So, is 7*5=35 the no. of isosceles triangles that can be drawn?

• No. You correctly found that for every vertex there are five isosceles triangles containing that vertex. However, for two different vertices $v, w$, the set of triangles containing $v$ and the set containing $w$ might overlap, so you can't say that the total will be 35. In fact, it happens that 35 counts the total number of possible triangles in your heptagon, isosceles or not. – Rick Decker Aug 23 '13 at 16:20

There are seven choices for the apex. For each apex there are three choices for the base line. Hence there are $21$ isosceles triangles. In total, there are $7\choose 3$ triangles, so the probability is $\frac{3}{5}$.

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Often on a weekend we take a drive to Welland (home of the ship canal) for a spot of shopping - typically feed for horses or poultry, and any other farm stuff we have forgotten (because TSC opens on Sunday unlike all the good christian folk nearer to home).

Across the road from TSC is a Value Village, where my wife loves to while away an hour or two spending \$30-80 on assorted kids clothes and nick-nacks. I get to park the car at the convenient charging station, and walk to the opposite end of the mall to join Steph in the store. Hopefully by that time she is more nearly done. I get 10 minutes or so to browse the electronics and books. This time I noticed a little keyboard - a logitach k-380. It seemed OK, but it didn’t have batteries so I couldn’t really test it. At \$5.99 I took the chance and we bought it. At home I put in a couple of AAA batteries, and paired it with my phone, and subsequently my Samsung Chromebook Plus. It works just fine. The best part is that my Chromebook Plus has been actting up in the keyboard department, having very stiff keys to the left end of the keyboard, and some keys would repeat constantly.

Using the bluetooth keyboard works around the issues completely and I can now type fluently once again. (I am typing this blog post using the new (used) keyboard.

I had in mind initially to use this keyboard for Raspberry Pi control - and I will still use it for that, but I foresee that I will use it the most for the Chromebook.


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# Possible count of solutions for clustering $n$ objects

Assuming $n$ objects are given. I like to get the number of possible solutions for a clustering algorithm. Obviously the cluster count must be greater or equal to $1$ and smaller or equal to $n$. For these two cases it is also easy to calculate how many solutions exists: For both exactly one.

If it is assumed there are $2$ clusters it is also not too hard: There are (probably) $2^n-2$ possible solutions.

I like to add up the amount of all solutions, but I fail to generalize the count of solutions for a specific amount of clusters (if this is given, then it should be easy).

Maybe someone knows how to do this?

Thank you very much

• Search for Stirling numbers of the second kind. – Phicar Jun 25 '17 at 21:57
• Cool, thanks:)! – Kevin Meier Jun 25 '17 at 22:09


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# what are the standard deviation and coefficient of variation of {5,6,7,9,8}?

Jun 28, 2016

Standard Deviation is $1.4142$ and Coefficient of variation is 20.2%

#### Explanation:

The mean $\left(\mu\right)$ of data set is given by the sum of data divided by their number i.e. $\frac{\Sigma x}{N}$

Hence mean is $\frac{1}{5} \left(5 + 6 + 7 + 9 + 8\right) = 7$

Standard Deviation $\left(\sigma\right)$ is given by sqrt[(Sigmax^2)/N-((Sigmax)/N)^2

$\frac{\Sigma {x}^{2}}{N} = \frac{1}{5} \left({5}^{2} + {6}^{2} + {7}^{2} + {9}^{2} + {8}^{2}\right)$

= $\frac{1}{5} \left(25 + 36 + 49 + 81 + 64\right) = \frac{255}{5} = 51$

Hence Standard Deviation is $\sqrt{51 - {\left(7\right)}^{2}} = \sqrt{2} = 1.4142$

Coefficient of variation is $\frac{\sigma}{\mu} \times 100$ = 1.4142/7xx100=20.2%


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WD transiting planets with LSST

# On the detectability of transiting planets orbiting white dwarfs using LSST

Jorge Cortés & David Kipping
Dept. of Astronomy, Columbia University, 550 W 120th Street, New York NY 10027
Accepted . Received ; in original form
###### Abstract

White dwarfs are one of the few types of stellar objects for which we know almost nothing about the possible existence of companion planets. Recent evidence for metal contaminated atmospheres, circumstellar debris disks and transiting planetary debris all indicate that planets may be likely. However, white dwarf transit surveys are challenging due to the intrinsic faintness of such objects, the short timescale of the transits and the low transit probabilities due to their compact radii. The Large Synoptic Survey Telescope (LSST) offers a remedy to these problems as a deep, half-sky survey with fast exposures encompassing approximately 10 million white dwarfs with apparent magnitude. We simulate LSST photometric observations of 3.5 million white dwarfs over a ten-year period and calculate the detectability of companion planets with  d via transits. We find typical detection rates in the range of to for Ceres-sized bodies to Earth-sized worlds, yielding to detections for a 100% occurrence rate of each. For terrestrial planets in the continuously habitable zone, we find detection rates of indicating that LSST would reveal hundreds of such worlds for occurrence rates in the range of 1% to 10%.

###### keywords:
eclipses — planets and satellites: detection — methods: statistical
pubyear: 2018pagerange: On the detectability of transiting planets orbiting white dwarfs using LSSTOn the detectability of transiting planets orbiting white dwarfs using LSST

## 1 Introduction

The vast majority of stars end their lives as white dwarfs (WDs). Within these stellar remnants, nuclear fusion has ceased and thus further inward collapse is resisted by electron degeneracy pressure. As a result, these remnants have compact radii similar to that of the Earth, yet with a mass of typically half of that of the Sun. Despite the absence of internal fusion, these stars shine for billions of years as they slowly cool, providing a means of studying their behavior and environment.

In recent years, there has been increasing interest in searching for and studying planets orbiting ever smaller primaries (Reich, 2013). In the early years of the modern exoplanet era, surveys typically focused on FGK stars resembling the Sun (e.g. Bakos et al. 2004; Pollacco et al. 2006; Bakos et al. 2013; Wilson et al. 2008). A basic argument was that such stars clearly have a credible chance of supporting life as demonstrated by our own existence, but smaller primaries (in particular M-dwarfs) may be less favorable111We note that more recent studies take a more optimstic view of M-dwarf habitability; for example see the review of Shields et al. 2016. (e.g. see Dole 1964; Kasting et al. 1993). Both radial velocity surveys, such as HARPS (Bonfils et al., 2013), and photometric surveys, such as MEarth (Charbonneau et al., 2009; Irwin et al., 2015), began to shift the focus towards smaller M-dwarfs, arguing that their smaller dimensions provide a significant boost to our sensitivity. The discovery by Kepler that early-type M-dwarfs appear to host more planets, including habitable-zone planets, than Sun-like stars (Dressing & Charbonneau, 2015) has brought M-dwarfs keenly into focus of planet hunters in recent years.

Surveys such as SPECULOOS (Burdanov et al., 2017) aim to push down further to late-type M-dwarfs as suitable targets for hunting planets. Surveys for planets orbiting even smaller and fainter brown dwarfs appear imminent (Triaud et al., 2013). Clearly then, the field of exoplanets has departed from the paradigm that we should only survey types of stars where we know for certain life is possible. There are certainly many challenges to life as we know it surviving and thriving on planets such as Proxima b (Anglada-Escudé et al., 2016) with a rich and active debate taking place in the literature (Ribas et al., 2016; Turbet et al., 2016; Garcia-Sage et al., 2017). Against this backdrop, an open-minded philosophy for allocating observational resources and effort is to focus on looking for life in places where we have the ability to observationally test it, not necessarily the places which we hypothesize as being the most habitable.

It could be argued that the pinnacle of this drive towards ever smaller planet hosts is represented by white dwarfs222since more compact objects are not luminous. With a radius of about ten times less than that of the smallest M-dwarf (Shipman, 1979; Chen & Kipping, 2017), these stars provide a major amplification of transit signals of up to 100-fold. On the downside, these stars are intrinsically faint, meaning that they are quite uncommon in a magnitude-limited survey. Further, their small dimensions mean that transits would last for minutes, not hours, posing a challenge to conventional surveys whose integration times are often too long to resolve the signals (Jenkins et al., 2010).

The potential value of a WD exoplanet survey was highlighted by Agol (2011), who argued that at least a few thousand WDs need to be surveyed to place meaningful constraints on their existence. The largest survey of WDs for transiting planets to date was recently published by Van Sluijs & Van Eylen (2017), who found no examples amongst WDs observed by K2. If such planets could be found, Loeb & Maoz (2013) estimate that only a few hours of JWST time would be needed to detect biosignatures in their atmospheres (should they exist) thanks to the small size of the host.

At first, it may seem a stretch to consider WDs as potential planet hosts. These stars must have passed through the red giant phase engulfing any planets within an AU (Sandquist et al., 1998, 2002). Despite this, there are now numerous indirect clues suggesting planets may indeed orbit WDs. First, % of WDs appear to have metal contaminated atmospheres, indicative of a continuous supply of in-falling rocky material as a result of the rapid convection times (Zuckerman et al., 2003, 2010; Koester et al., 2014). Second, debris disks appear common around WDs, with a lower limit being that 5% host such disks potentially supporting a second generation of planet formation (Barber et al., 2012). Third, there is direct evidence of a likely disintegrating planet orbiting WD 1145+017 (Vanderburg et al., 2015). Put together, these clues strongly motivate that we should at least attempt a deep search for planets orbiting white dwarfs.

As alluded to earlier though, there are two significant obstacles facing any survey attempting to seek WD planets. First, WDs are faint and thus in order to survey a large number we need to survey both a large fraction of the sky and go deep (Kilic et al., 2013). Second, the transits last for minutes, meaning that exposures must not exceed that timescale in order to avoid significant distortion and dilution of the transit morphology (Kipping, 2010). LSST is essentially completely unique in being able to overcome these two challenges. The sample size issue will be certainly overcome, since LSST will observe down to magnitude, including an expected WDs (Agol, 2011). The problem of transit distortion is also overcome since LSST is expected to take two 15 second exposures back to back in normal operation, sufficient to avoid smearing of the light curve (Abell et al., 2009).

The most obvious drawback in using LSST for this purpose is that it does not survey each patch of the sky for long periods of time. Thus, one should never reasonably expect a particular instance of WD transit to be covered, only a partial transit (with the exception of the deep drilling fields). However, if the signal is strictly periodic, then it is not the temporal coverage which we actually care about but rather the phase coverage (Lund et al., 2015). Over the ten year baseline of LSST we should expect well-suited phase coverage for any given star333This has been previously demonstrated to certainly be true for transiting planets of normal stars, see Jacklin et al. (2015, 2017)..

For the reasons described above, we hypothesized that LSST would be an excellent machine for searching for WD transiting planets. However, a detailed study injecting real transit signals into LSST-like sparsely sampled time series around realistic WDs is notably absent in the literature and thus reasonable concerns might exist about the true feasibility of detecting WD planets with LSST. In response to this, the work presented in what follows offers a detailed suite of simulations of planets injected into LSST light curves to evaluate their detectability.

## 2 Simulating LSST Light Curves

### 2.1 Overview: A Monte Carlo Approach

The primary objective of this work is to evaluate the detectability of transiting planets with LSST. An unconditional yield estimate is not formally calculable in the absence of any information about the occurrence rate and distribution of planets orbiting WDs. Instead, we aim to ask the question, if a WD hosted a planet with radius and orbital period , how detectable would that planet be using LSST444We return to the issue of yield estimation later in Section 3, where we show that a yield estimate can be estimated conditioned upon specific assumptions about the underlying distribution of WD planets.?

This question is tackled from a Monte Carlo perspective using numerical simulations. Whilst it may certainly be possible to express a reasonable parametric model describing the detectability of WD planets using analytic arguments, it is clearly complicated by the sparse non-uniform scheduling expected with LSST (Abell et al., 2009). A Monte Carlo approach is attractive if there exists a means of generating representative photometric time series expected from LSST for WDs. Unpacking that statement, the requirements of such an approach can be more specifically stated as being a) the need to simulate a representative distribution of the properties of WDs that will be observed by LSST b) the need to simulate representative photometric time series expected of said stars, accounting for realistic LSST noise and scheduling constraints.

Fortunately, both of these requirements are satisfied by software resources made available by the LSST team, namely OpSim (Coffey et al., 2006) and CatSim. Using these tools then, we describe in what follows how we generate representative photometric time series of WDs that will be observed by LSST. We discuss our approach for measuring planet detectability later in Section 3.

### 2.2 Generating a WD Catalog with CatSim

LSST’s Catalog Simulator (CatSim) is used to incorporate a realistic distribution of WDs within the Milky Way Galaxy. WDs, and all stars accessed by CatSim  are generated by galfast, a GPU-accelerated package that fits models of a thick and thin disk, and halo to SDSS data (Juric et al., 2008). The default simulated universe accessed by the CatSim stack is stored as a database on fatboy, a machine located at the University of Washington. Information for all WD’s with K was retrieved and stored locally. The temperature constraint is imposed in consideration of the white dwarf habitable zone (Agol, 2011).

### 2.3 Interpolating Stellar Properties

Stars listed in the fatboy database state the and of each source. However, in order to inject a planet around a given WD, we need to know the stellar mass, , allowing us to convert a chosen orbital period into semi-major axis via Kepler’s Third Law555We note that a stellar density would also suffice, but this is also unavailable.. Rather than simply adopt a uniform stellar density for all WDs, we seek to create the most realistic catalog possible in this work. Accordingly, we elected to estimate a realistic stellar mass for each WD based on the provided CatSim information.

To accomplish this, we used the evolutionary cooling models of hydrogen-and helium-atmosphere white dwarfs666Available at this URL from Holberg & Bergeron (2006); Kowalski & Saumon (2015); Tremblay et al. (2011); Bergeron et al. (2011). We perform a bi-linear interpolation of these model grids such that for any combination of and , we are able to assign a unique stellar mass. We find that both the H-rich & He-rich WDs masses approximately follow a normal distribution peaked at half a solar mass with a standard deviation.

### 2.4 Generating Light Curves with OpSim and CatSim

To generate mock observations, LSST’s Operations Simulator (OpSim) is used in conjunction with CatSim777Tutorial notebooks from LSST available at this URL.. OpSim provides a sample cadence for LSST’s ten-year observing strategy; we make use of the current reference run, minion_1016. As it currently stands, the CatSim stack will only produce light curves for objects that incorporate a variability model; thus, we set the variability constraint within LightCurveGenerator.py of CatSim to ‘None’. At this point, we are able to retrieve light curve data (time, magnitude, errors) for all WDs.

To avoid memory issues on our local machine, we excluded data for WDs within of galactic longitude, as seen in Figure 1. Additionally, this constraint has the added benefit of exploring regions which are less affected by crowding888Whilst crowding is not an issue for simulations, such as those used here, crowded fields may lead to source contamination during the actual LSST observations.. All WD light curves were stored locally on a database file.

We briefly point out that the light curves generated assume that the visit-to-visit calibration error is much less than that of the point-to-point photometric uncertainty. Given our sources are generally faint, the point-to-point uncertainties are already large meaning that if LSST achieves visit-to-visit calibrations of order of a percent or better, this is unlikely to be a meaningful source of error999Unlike the case of Sun-like stars where likely sub-mmag long-term calibration would be needed.

## 3 Planet Injection & Recovery

### 3.1 Overview

In order to calculate the detectability of planets around WDs with LSST, it is of course necessary to inject planets into our synthetic light curves described in Section 2, and so we turn our attention to this here.

As touched on earlier, the distribution and occurrence rate of planets around WDs is broadly unknown, making yield estimates, at best, conditional (we define what we mean by conditional in detail shortly). Accordingly, we place our emphasis here on estimating the marginalized detectability of a planet of a given size and period. The process of marginalization is discussed in Section 3.4.

### 3.2 Injecting planets

Using a random subset of 3.5 million of our simulated WD light curves, we inject a single planet around every star with a random orbital period and physical radius. Periods are drawn from a log-uniform distribution between 0.15 d and 10 d, and radii from a log-uniform distribution from to . A random impact parameter, b, between and is assigned to each planet, such that a transit is guaranteed. The time of inferior conjunction, , is randomly assigned for each, and every planet is assumed to follow a strictly circular Keplerian orbit.

Light curves for the injected planets are simulated using batman (Kreidberg, 2015) assuming a 15 s integration time and uniform limb darkening. In all cases, we assume that the eventual ten-year time series is available for the analysis. Further, we also highlight that we assume only a single planet for each star and that 100% of the stars have planets (it is straightforward to scale our results for arbitrary occurrence rates later).

For each star, we calculate the planet’s signal-to-noise ratio (S/N) and assign a yes/no binary flag as to whether LSST is deemed to be sensitive to said planet (using the method described in Section 3.3). Accordingly, amongst the 3.5 million stars, we can select those that have planets within a local size- and period-bandwidth of a particular choice of and , and then simply count up what fraction of the stars had planets that LSST was sensitive to. This would therefore represent the marginalized sensitivity. This exercise could be repeated but instead using only a subset of the 3.5 million stars - for example taking just the bright end. In this way, the marginalized sensitivity can be computed using whatever marginalized sample one desires.

### 3.3 S/N and sensitivity

In this work, we inject planets but we do not blindly recover them. The sensitivity of LSST to a given planet is computed by evaluating the S/N instead. This choice was largely motivated by computational practicality. Running a box least squares blind search (Kovács et al., 2002) on 3.5 million light curves would represent a major computational challenge.

We define S/N as follows. If the data can be assumed to approximately follow a diagonal multivariate normal distribution, then one may define the goodness-of-fit of a specific model using Pearson’s chi squared. Accordingly, we compare the of a null flat line model through the data versus that of the planet model. We assume that any long-term variability has been filtered (e.g. with a high-pass filter) and that short-term variability is much lower amplitude than the transit signals injected (which often approach a 100% eclipse depth). For our test, we use the exact parameters for the planet model as that used for the injection, and thus this is why our approach here is certainly not a blind recovery. However, the difference between these two merit functions can be used to define S/N as follows:

 S/N ≡√χ2null−χ2planet. (1)

In order to assess whether LSST is deemed to be sensitive or non-sensitive to a particular planet, we use a simple S/N threshold. Therefore, signals with a S/N exceeding the threshold are always defined as LSST-sensitive, and otherwise insensitive. This is a simplifying assumption since real transit surveys do not have step functions sensitivity curves at a particular threshold but rather S-like curves centered around a certain value (e.g. see Christiansen et al. 2016). Nevertheless, certainly recoverability does saturate to unity beyond a certain point and thus a threshold is not an unreasonable approximation. Ultimately, the true curve will not be known until real data becomes available. We adopt a S/N threshold of 7.1 in what follows, as this was the same value initially adopted by the Kepler team (Jenkins et al., 2010) and thus provides a standardized point of comparison.

In practice, the S/N is computed using all of the available LSST bandpasses in conjunction to maximize our sensitivity to transit signals. An example of this is shown in Figure 2 for illustration.

### 3.4 Sensitivity in period-radius plane

After the calculations are complete, we have assigned a yes/no detection flag to a total of 3.5 million individual simulated systems of randomized period and radius. The sensitivity of LSST to each system is clearly quite varied and depends on the star’s magnitude, position, observing cadence, etc. To simplify the picture, we calculate a so-called marginalized sensitivity as a function of and .

This is accomplished by first defining a local size- and period-bandwidth and then moving across a two-dimensional grid of and and calculating the local sensitivity as the the number of positives in that window divided by the total stars in that window. In this way, the estimate has marginalized (or averaged) over all other parameters, such as stellar properties.

We define a bandwidth such that our periods and radii are divided into evenly spaced grid points in logarithmic space, which we found provides a good balance of sufficient numbers per bin as well sufficient number of bins. The resulting marginalized sensitivities are plotted in Figure 3 and are made publicly available at this URL. In our grid, it is apparent that we have removed some of the shortest period objects from our sample. These censored objects fall within the Roche limit of the star, where we have converted planetary radii into masses and then densities using the forecaster empirical mass-radius relation (Chen & Kipping, 2017).

It is important to remember that sensitivity does not account for transit probability and thus one might reasonably expect sensitivities approaching unity for optimal cases. Indeed, the dynamic range apparent in Figure 3 reflects this. As expected, short-period planets are evidently more easily detected than their longer period bretherin due to the increased frequency of their transits. The strongest bias occurs along the radius axis, where naturally larger planets are much more easily detected.

Sensitivity has a dynamic range from zero to unity. It should be expected to saturate to unity in an exponential manner for highly idealized cases. Similarly, it should be expected to saturate to zero for extremely challenging cases. We therefore considered that sensitivity likely follows a logistic function. Accordingly, let us take a single slice along the log-period axis for a fixed choice of planetary size (). We would expect sensitivity along the log-period slice to be described by

 S =11−kexp(log(P/days)−logP0), (2)

where is a free parameter quantifying the steepness of the logistic curve and is a free parameter defining the mid-point.

We regressed this expression along all available choices of (removing cases where no detections were found), and found that is consistent across all choices with a mean and standard deviation of . In contrast, the term appears to linearly increase with respect to . If we replace in Equation 2 with a straight-line slope with respect to , the result may be rearranged to the form,

 S =11+a(P/days)b(RP/R⊕)c, (3)

where we find the values , and provide an excellent fit. The positive value for indicates that longer period planets are more difficult to detect, close to a dependency. The reason why the scaling is better than , which one would expect if considering purely the transit frequency scaling, is due to the effect of increased durations at longer being preferentially detectable.

The negative coefficient indicates a roughly linear scaling of sensitivity with respect to planet size. The relationship is not quadratic, as one might naively expect, due to the fact that most of the detectable region of our parameter space occurs for , where quadratic scaling is not expected due to the total-eclipse and grazing-nature of the transits which dominate.

We highlight that directly comparing these coefficients to the theoretical expectations (e.g. from Kipping & Sandford 2016) is not generally possible since conventional transit yield/bias calculations do not operate in a regime dominated by grazing configurations. Nevertheless, Equation 3 with the quoted coefficients provides a straightforward way for the community to use our sensitivity results in other studies.

### 3.5 From sensitivity to detectability

So far, our discussion has focused on sensitivity, which is defined under the assumption that the impact parameter is uniformly distributed between zero and . Thus, all stars are assumed to have a transiting planet. Of course, even if all of the stars have planets, they will not all host transiting planets. We therefore define marginalized detectability as being similar to marginalized sensitivity except we now account for the geometric transit probability expected for each planet.

In the case of sensitivity, injected planets were either flagged as detected (yes) or or non-detected (no). To compute detectability, we multiply these essentially binary probabilities by the geometric probability of , which equals . Once each planet has a detection probability assigned, we draw a random Bernouilli integer to define the injected planet as being detected or non-detected 101010We could have also used the probabilities themselves for the subsequent marginalization, but we elected to binarize the results such that they more closely resemble a real survey.. We are then able to compute a marginalized detectability in a similar fashion to that described earlier for sensitivity.

### 3.6 Detectability in period-radius plane

Our detectability results, accounting now for transit probability, are illustrated in Figure 4 and are also made available at this URL. Broadly speaking, the pattern appear similar to that of sensitivity, except the resulting scores are typically two orders of magnitude lower, reflecting the % transit probability of our injected planets. The dynamic range increases due to longer period planets being particularly unlikely to transit.

The mean detectability across all simulations equals 0.107%, implying that roughly in WD systems harboring planets need to be surveyed by LSST to bag a single detection. We find that, marginalized across radius, the mean detectability drops off as , peaking at 1.5% for the shortest Roche-stable period possible ( of a day) and dropping down to 0.025% at  d.

Using these numbers, it is straightforward to estimate yields for various occurrence rates. If 100% of the WD stars observed by LSST harbor a planet with  d (i.e. ) then we would expect detections. Thus, we would require in order for LSST to detect no examples of transiting WD planets. On this basis, an LSST survey for WD transiting planets would be highly informative, either delivering many examples of a new class of planetary system or demonstrating to high confidence such systems are rare.

Although the log-uniform distributions in period and radius are likely as good as a choice as any other, the more arbitrary choice in our prior is the minimum/maximum period/radius. Changing these bounds will strongly influence a marginalized yield estimate, such as that made above. Rather than go through various hypothetical and speculative scenarios for these bounds, we prefer to focus on the more robust kernel density estimates for detectability (i.e. those shown in Figure 4), which highlight that amongst a sample of WDs, the vast majority of parameter space should be expected to yield detections.

### 3.7 Detection bias

Armed with our numerical simulations, it is possible to investigate the impact of detection bias on our results. White dwarfs were generated as a representative astrophysical population using CatSim, but planet detections will clearly favor brighter stars and bigger planets. Figure 5 compares the detected vs injected population for four key parameters. We find that the stellar masses and radii of detected cases are representative of the true population. However, the detected stars tend to be hotter and thus more luminous making their photometric time series better quality for planet detections. The strong and expected bias towards larger planets is also recovered.

### 3.8 Detectability of temperate transiting WD planets

A strong motivation behind the search for planets outside of our solar system is the potential characterization of habitable worlds, we thus turn our attention to a WD’s temperate zone here. As discussed in Agol (2011), given a WD’s mass and type (either H-rich or He-rich), a planet requires a specific orbital distance to fall within the continuously habitable zone (CHZ). This is defined as the range of orbital radii for which a planet will receive the necessary flux to sustain liquid water on its surface for at least 3 Gyr. We take the outer limits of the CHZ for a H-rich and He-rich WD from Agol (2011) and reassess the detectability results shown in Figure 4. We note that the inner limit of the CHZ comes up against the tidal disruption limit and thus can essentially be ignored in what follows.

We re-cast our detection figure in terms of WD mass and semi-major axis, so we can directly draw the CHZ contours from Agol (2011) on top. In this way, the x-axis is essentially re-scaled via Kepler’s Third Law from the version in Figure 4. The y-axis has replaced planetary radius with stellar mass, which was previously a marginalized quantity. It is therefore clear that in the revised version here, planetary radius will be a marginalized quantity. Specifically, we marginalize planets in the range to focus on the potentially terrestrial-like planets. Finally, we split our sample into H-rich and He-rich stars, since the CHZ is noticeably different between the two (Agol, 2011). The final results are shown in Figure 6.

We briefly remark that the detectability rates are in the range of to , with the inner edge becoming attenuated due the impact of tidal disruption. This suggests that if the occurrence rate of CHZ rocky planets around WDs is , we should anticipate detections with LSST.

## 4 Discussion

We have demonstrated that LSST will have the capability to detect transiting planets around white dwarfs. Over an assumed ten year baseline, the sporadic sampling of LSST combines together to provide the excellent phase coverage needed for succesful transit detection (also see Lund et al. 2015; Jacklin et al. 2015, 2017). Further more, LSST’s short integration time of 15 seconds ensures that WD transits, which typically last for a minute or two, are not significantly distorted due to binning (Kipping, 2010) aiding their identification. These two advantages enable detection but it is LSST’s incredible depth, exceeding magnitude, which make LSST potentially a revolution in the quest to detect WD planets since the survey will observe WDs.

We find that detection rates for  d planets range from for long-period Ceres-sized bodies to for short-period Jovians. Rates are naturally highly dependent upon what type of planet is under consideration but our suite of results are made publicly available at this URL. Yield estimates are not directly possible due to our lack of information about the WD planet population. However, as an example we highlight that if % of WDs host a Mars-sized planet with  d, we should expect detections with LSST, and thus one might reasonable expect hundreds of discoveries.

If terrestrial planets reside in the continuous habitable zones of WDs with a frequency greater than , LSST should be expected to detect examples. Assuming again % would yield detections. We would expect the brightest planet hosting WD in such an example to be - and thus may be suitable for atmospheric and even biosignature characterization with JWST (Loeb & Maoz, 2013).

We highlight some limitations of our study. First, our work assumes that the transit detection pipeline acts as a perfect detector for and thus we did not execute blind recoveries of injected signals. Although typical search algorithms are found to have high efficiencies in this regime (e.g. see Christiansen et al. 2016), white dwarfs have not been surveyed in great detail before. Second, we assume long-term photometric behavior is filterable and that short-term variations have an amplitude less than the formal uncertainties. Given the faintness of our targets, we argue this is a reasonable approximation where photon noise dominates. Finally, we highlight that whether a planet in the CHZ is truly habitable is a completely different question that we make no attempt to investigate in this work.

Ultimately, our work makes a strong case that although LSST may not have been built with WD transits as a science case in mind, it is uniquely placed to conduct the most in-depth survey to date. A preliminary survey with K2 of 1148 WDs by Van Sluijis & Van Eylen (2018) found no transiting objects and placed upper limits on planet occurrence rates in the range of to for  h to 10 d roughly Earth-sized bodies. For comparison, if LSST failed to detect similar objects, the occurrence rate would be constrained to be %. With prior indications of planetary material falling onto % of WDs (Zuckerman et al., 2003, 2010; Koester et al., 2014), a search for minor bodies around such stars is both timely and critical for advancing our understanding of these intriguing environments.

## Acknowledgments

DMK is supported by the Alfred P. Sloan Foundation. JC gratefully acknowledges the support of the Columbia University Bridge to Ph.D. Program in the Natural Sciences and the support provided by the NSF through grant AST-1539931. Special thanks to Scott Daniel for his guidance with using CatSim and to Jay Farihi for his helpful comments on this work. Thanks to the members of the Cool Worlds Lab for regular discussions and feedback during the preparation of this work.

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 Title: Angle resolved photoemission studies of silver-bromide Author(s): Kwawer, Gary Neil Doctoral Committee Chair(s): Brown, Frederick C. Department / Program: Chemical Physics Discipline: Chemical Physics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Physics, Condensed Matter Abstract: In this work, polarized synchrotron radiation is used to obtain both angle resolved photoemission data and angle integrated photoemission data on randomly oriented thick films of metallic silver, thin films of randomly oriented polycrystalline AgBr and AgBrI, and epitaxially grown AgBr on an Ag (111) single crystal.The experiments on the randomly oriented samples were concerned with obtaining the correct sample preparation methods--evaporation of AgBr or AgBrI onto Ag which was evaporated onto an Au plated glass disc; the correct temperature conditions--sample temperature must be kept below 140K, preferably close to 77K, in order to stop sample photolysis and decrease electron-phonon interactions. Asymmetry parameters as well as relative cross sections were determined for the valence bands of Ag and AgBr from 55eV through the Ag 4d Cooper minimum at 140eV. Comparison of asymmetry parameters with atomic data indicates that the valence bands derived from the Ag 4d levels in these systems are remarkably atomic-like. In AgBr, only the uppermost valence bands show a greatly reduced excursion of $\beta$-value through the Cooper minimum.The total and partial valence band density of states (DOS and PDOS) for AgBr and AgBrI were experimentally obtained. The Ag 4d PDOS was nearly identical for both compounds. The halogen derived PDOS's differed in that the AgBrI halogen PDOS extended 0.2eV closer to the vacuum level than its AgBr counterpart.Experiments on AgBr epitaxially grown on Ag (111) single crystals demonstrated a definite surface core level shift (SCS) towards lower binding energy by 0.8eV for the Br 3d subshell. Unfortunately, these results do not conclusively state which ion terminates the surface. The general belief is that an AgBr (111) surface is bromide terminated.Preliminary band mapping experiments were performed on epitaxially grown AgBr on an AgBr (111) single crystal from $\Gamma$ to L in the first Brillouin zone which experimentally located the valence band maximum at L. Comparing the experimentally obtained band structure with speculative ones shows that the experimental ones are flatter than their speculative counterparts. Issue Date: 1989 Type: Text Language: English URI: http://hdl.handle.net/2142/21830 Rights Information: Copyright 1989 Kwawer, Gary Neil Date Available in IDEALS: 2011-05-07 Identifier in Online Catalog: AAI8916273 OCLC Identifier: (UMI)AAI8916273


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# $(a-b)^2$ Formula

## Formula

$(a-b)^2 \,=\, a^2+b^2-2ab$

### Introduction

Let $a$ and $b$ represent two terms in algebraic form. The difference of them is equal to $a-b$, which is a binomial and the square of the binomial $a-b$ is expressed as $(a-b)^2$ in mathematical form. It is also known as the square of the difference of the two terms.

The $a-b$ whole square is used as a formula to expand it as an algebraic expression $a^2-2ab+b^2$ in mathematics.

$\implies$ $(a-b)^2 \,=\, a^2+b^2-2ab$

##### Usage

The square of the difference of two terms formula is used in mathematics in two different cases.

1. It is used to expand the square of difference of two terms or a binomial.
2. If a mathematical expression is in the form of $a^2+b^2-2ab$, then it is simply written as $(a-b)^2$.

#### Examples

$(1) \,\,\,$ Expand $(2x-5y)^2$

Take $a = 2x$ and $b = 5y$, and replace them in the expansion of the formula for calculating the value in mathematical form.

$\implies$ $(2x-5y)^2$ $\,=\,$ $(2x)^2+(5y)^2-2(2x)(5y)$

$\implies$ $(2x-5y)^2$ $\,=\,$ $4x^2+25y^2-2 \times 2x \times 5y$

$\,\,\, \therefore \,\,\,\,\,\,$ $(2x-5y)^2$ $\,=\,$ $4x^2+25y^2-20xy$

$(2) \,\,\,$ Simplify $9m^2+16n^2-24mn$

$\implies$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m)^2+(4n)^2-24mn$

$\implies$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m)^2+(4n)^2-2 \times 3 \times 4 \times m \times n$

$\implies$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m)^2+(4n)^2-2 \times 3 \times m \times 4 \times n$

$\implies$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m)^2+(4n)^2-2 \times (3m) \times (4n)$

$\implies$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m)^2+(4n)^2-2(3m)(4n)$

Take $a = 3m$ and $b = 4n$. Now, simplify the algebraic expression by the $(a-b)^2$ algebraic identity

$\,\,\, \therefore \,\,\,\,\,\,$ $9m^2+16n^2-24mn$ $\,=\,$ $(3m-4n)^2$

### Proofs

The $a$ minus $b$ whole squared identity can be derived mathematically in two different methods.

1. Learn how to derive the expansion of the $a-b$ whole square formula by the multiplication in algebraic approach.

2. Learn how to derive the expansion of the $a-b$ whole squared identity by the areas of geometric shapes.

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###### Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.


text:
# Sampling methods¶

Capabilities for sampling of random input parameters.

This module contains functions to sample random input parameters.

econsa.sampling.cond_mvn(mean, cov, dependent_ind, given_ind=None, given_value=None, check_cov=True)[source]

Conditional mean and variance function.

This function provides the conditional mean and variance-covariance matrix of [$$Y$$ given $$X$$], where $$Z = (X,Y)$$ is the fully-joint multivariate normal distribution with mean equal to mean and covariance matrix cov.

This is a translation of the main function of R package condMVNorm.

Parameters
• mean (array_like) – The mean vector of the multivariate normal distribution.

• cov (array_like) – Symmetric and positive-definite covariance matrix of the multivariate normal distribution.

• dependent_ind (int or array_like) – The indices of dependent variables.

• given_ind (array_like, optional) – The indices of independent variables (default value is None). If not specified return unconditional values.

• given_value (array_like, optional) – The conditioning values (default value is None). Should be the same length as given_ind, otherwise throw an error.

• check_cov (bool, optional) – Check that cov is symmetric, and all eigenvalue is positive (default value is True).

Returns

• cond_mean (array_like) – The conditional mean of dependent variables.

• cond_cov (arrray_like) – The conditional covariance matrix of dependent variables.

Examples

>>> mean = np.array([1, 1, 1])
>>> cov = np.array([[4.0677098, -0.9620331, 0.9897267],
...                   [-0.9620331, 2.2775449, 0.7475968],
...                   [0.9897267, 0.7475968, 0.7336631]])
>>> dependent_ind = [0, ]
>>> given_ind = [1, 2]
>>> given_value = [1, -1]
>>> cond_mean, cond_cov = cond_mvn(mean, cov, dependent_ind, given_ind, given_value)
>>> np.testing.assert_almost_equal(cond_mean, -4.347531, decimal=6)
>>> np.testing.assert_almost_equal(cond_cov, 0.170718, decimal=6)

Conditional sampling from Gaussian copula.

This module contains functions to sample random input parameters from a Gaussian copula.

econsa.copula.cond_gaussian_copula(cov, dependent_ind, given_ind, given_value_u, size=1)[source]

Conditional sampling from Gaussian copula.

This function provides the probability distribution of conditional sample drawn from a Gaussian copula, given covariance matrix and a uniform random vector.

Parameters
• cov (array_like) – Covariance matrix of the desired sample.

• dependent_ind (int or array_like) – The indices of dependent variables.

• given_ind (array_like) – The indices of independent variables.

• given_value_u (array_like) – The given random vector ($$u$$) that is uniformly distributed between 0 and 1.

• size (int) – Number of draws from the conditional distribution. (default value is 1)

Returns

cond_quan – The conditional sample ($$G(u)$$) that is between 0 and 1, and has the same length as dependent_ind.

Return type

numpy.ndarray

Examples

>>> np.random.seed(123)
>>> cov = np.array([[ 3.290887,  0.465004, -3.411841],
...                 [ 0.465004,  3.962172, -0.574745],
...                 [-3.411841, -0.574745,  4.063252]])
>>> dependent_ind = 2
>>> given_ind = [0, 1]
>>> given_value_u = [0.0596779, 0.39804426]
>>> condi_value_u = cond_gaussian_copula(cov, dependent_ind, given_ind, given_value_u)
>>> np.testing.assert_almost_equal(condi_value_u[0], 0.856504, decimal=6)


text:
main-content

## Weitere Artikel dieser Ausgabe durch Wischen aufrufen

21.06.2019 | Polymers & biopolymers | Ausgabe 19/2019 Open Access

# Tailored polyurethane acrylate blend for large-scale and high-performance micropatterned dry adhesives

Zeitschrift:
Journal of Materials Science > Ausgabe 19/2019
Autoren:
Dan Yu, René Hensel, Dirk Beckelmann, Michael Opsölder, Bruno Schäfer, Karsten Moh, Peter William de Oliveira, Eduard Arzt
Wichtige Hinweise

## Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Introduction

Geckoes and some species of insects and spiders show superior dry adhesion due to their hairy attachment pads which enable them to locomote on various surfaces [15]. Mimicking those by transferring the underlying “contact splitting” principle into synthetic adhesives exhibits large potential for next generation adhesives [3, 6, 7]. Several studies have reported that splitting a nominal flat adhesive surface into several discrete contacts improved their adhesion performance; the reasons are related to scaling effects, enhanced compliance of the contact and a higher defect tolerance in combination with crack trapping mechanisms, which dissipate stored strain energy and prevent uncontrolled detachment [716]. Further work has focused on the optimization of design parameters such as tip geometry, aspect ratio of the microstructures, areal density [11, 17, 18] and mechanical properties such as (visco)elasticity and stiffness gradients [1923]. In addition, counter surface roughness [24], substrate elasticity [9], temperature [25] and humidity [26] further impact the performance and have to be taken into account in real-world applications.
Despite great progress in understanding the underlying concepts and the successful fabrication on a laboratory scale [17, 2729], the transfer to large-scale or even continuous manufacturing remains challenging due to restrictions in materials selection and due to the complex shape of the required microstructures [7]. Most critically, a continuous fabrication process requires materials with short curing times. Thus, light-induced cross-linking reactions are preferred, compared to thermally induced reactions, which often require longer heating times. Acrylate- or methacrylate-based polymers are materials with tunable viscoelasticity and toughness suitable for generation of micropatterned functional surfaces [3037]. By adding photo-cleavable entities (i.e., photoinitiators), radical-based cross-linking reactions can be induced by exposure to light. In addition, their properties can be tailored by tuning the chemical composition of the pre-polymers, which allows customizing for specific demands. In general, (meth)acrylate-based polymer blends contain an oligomer with (meth)acryloxy group, a photoinitiator and several modulators, which control properties such as viscosity of the pre-polymer [3032], surface free energy [30, 33, 38, 39], permeability [39] and curing kinetics. Mechanical properties can be tuned by the amount of functional entities (i.e., (meth)acryloxy group), the molecular weight of the oligomer and the chemical structure of the modulators, i.e., cycloaliphatic or linear chains [30, 3234, 40]. Customized (meth)acrylate-based materials have been reported as mold materials for replica molding as well as materials for generating micropatterned surfaces by a wide range of lithographic techniques [28, 33, 41, 42]. For example, Lee et al. reported on mushroom-shaped dry adhesives made from polyurethane acrylate (PUA) in a continuous roll-to-roll process. They took advantage of incomplete curing during fabrication of micropillars and added the mushroom tips by a subsequent compression and final curing treatment [43]. Another study by Yi et al. [34] demonstrated a roll-to-roll process for polyurethane acrylate materials, but it was limited to transparent templates.
We recently reported a one-step process for continuous fabrication of high-performance mushroom-shaped dry adhesives on large-scale roll-to-roll system, featuring a total web length of 24 m and a steel imprinting roll of 50 cm in diameter, to simulate an industrial fabrication process [44]. We found that the complex geometry including undercuts of mushroom-shaped adhesives can be replicated into polyurethane-acrylate (PUA)-based materials by using flexible, non-transparent templates and controlled compressive loads during the imprint process. In adhesion tests, we identified the backing layer thickness as a parameter relevant for the adhesion performance. Briefly, for too thin backing layers, the adhesion strength of the microstructures and their interfacial bonding to the polyethylene terephthalate (PET) carrier film decreased or was lost.
In the present work, we will show that the processing window can be expanded by introducing a double-layer coating strategy. Furthermore, we will provide detailed process parameters for precise replication of mushroom tips and will correlate morphology with adhesion results. Issues of replicating microstructures by using UV-curable materials will be discussed from aspects of material compositions: We varied the chemical composition of the PUA-based microstructures by blending with additives to optimize the processability in a continuous fabrication and the resulting adhesion properties. Additives to control UV-curing kinetics, the formation of mushroom caps and the influence of the mechanical properties of the materials were investigated. As a result, a formulation for improved adhesive mushroom-shaped microstructures, with dimensions replicating accurately those of the Ni-shim master structures, and for achieving optimal adhesion characteristics will be presented.

## Experimental section

### Material preparation

For the micropatterned dry adhesives, pre-polymer blends based on aliphatic urethane diacrylate oligomers, UA16 (Miramer UA5216, Miwon Specialty Chemical Co. Ltd., Gwanggyo, South Korea) were used. The photoinitiator Omnirad 500 (IGM Resins B.V., Waalwijk, The Netherlands) was added to all UA16-based blends in a concentration of 5 wt%. For different formulations, modulators were added: U6L7 contained 10 wt% Laromer LR 8887 (BASF AG, Ludwigshafen, Germany); U6S9 contained 1 wt% Miramer SIP900 (Miwon Co.); U6E4 contained 20 wt% Miramer ES4420 (Miwon Co.). All compositions for each pre-polymer blend are summarized in Table 1. All blends were mixed at 2350 rpm and degassed at 1 mbar for 3 min using a SpeedMixer (DAC600.2 VAC-P, Hauschild & Co. KG, Hamm, Germany).
Table 1
Compositions and materials properties of the adhesives and the flexible templates
Material
Descriptiona
Viscosity at 25 °C (Pa s)
Elastic modulus (MPa)b
Elongation at break (%)c
Surface free energy (mJ m−2)
UA16
Miramer UA5216
13.7
359
326
40.3
U6L7
UA16 + 10% Laromer 8887
8.2
243
204
29.2
U6S9
UA16 + 1% Miramer SIP900
13.9
357
196
24.6
U6E4
UA16 + 20% Miramer ES4420
10.6
150
448
24.3
nPDMS
Non-transparent PDMS
0.76
700
25.8
tPDMS
Transparent PDMS
1.64
400
24.6
aAll the UA16 blends (UA16, U6L7, U6S9, U6E4) included 5 wt  % Omnirad 500 (photoinitiator)
bRepresents storage modulus at room temperature obtained by DMTA.
cMeasured by tensile test
For the flexible templates, two types of silicones were used: Elastosil M4601 (nPDMS, Wacker Chemie AG, München, Germany) and Elastosil M4644 (tPDMS, Wacker AG). The two-component polydimethylsiloxanes were mixed at 2350 rpm and degassed at 1 mbar for 3 min using a SpeedMixer. The pre-polymers were cast into a circular mold (diameter 130 mm; depth 5 mm), in which the master structure was mounted. A micropatterned Ni-shim (Temicon GmbH, Dortmund, Germany) was used as a master structure that comprised mushroom-shape micropillars with diameters and heights of 50 µm. The diameters of the mushroom tips were in the range of 66–76 µm. The pillar array was hexagonally arranged with a center-to-center distance of 100 µm. The mushroom tips hence cover an area fraction of about 42.9%. The PDMS pre-polymer was then cured at 70 °C for 1 h, and the negative microstructure was received after gentle demolding. Several PDMS stamps were prepared by repeating this procedure and then trimmed into square shape, and eventually glued onto the imprinting roll.

### Materials characterization

Viscosities of the pre-polymer blends were measured with a rheometer (Physica MCR300, Anton Paar GmbH, Graz, Austria) in the temperature range between 25 and 80 °C at a constant shear rate of 1 s−1. For all materials, dynamic mechanic thermal analyses (DMTA) were performed in the temperature range from − 100 to 120 °C in tensile mode at an oscillatory frequency of 1 Hz under N2 atmosphere. The sample dimensions were 30 mm × 5.5 mm × 2 mm. Elongations at break were measured by (quasi-static) tensile tests (Material Testing Machine Xforce P, Zwick & Roell GmbH & Co. KG, Ulm, Germany) with a crosshead moving rate of 20 mm min−1. Dog-bone-shaped specimens with a thickness of 3 mm exhibited a gage length of 60 and a width of 8 mm. The width of the grips was 17 mm. For all mechanical tests, UA16 blends were cured by UV exposure (365 nm) for 5 min; the silicones were thermally cured at 70 °C for 1 h.
Surface free energies were examined by contact angle goniometry (OCA 35, analysis software: SCA20; DataPhysics Instruments GmbH, Filderstadt, Germany) using water and hexadecane as testing liquids. Smooth films with 120 µm thickness for static contact angle measurements were prepared by coating UA16 polymer blends or PDMS pre-polymers onto flat PET substrates, following by 5-min UV exposure for UA16 polymer blends and thermal curing at 70 °C for PDMS. The surface free energy was deducted from the obtained contact angles by using Wu’s harmonic method [45, 46].
UV-curing rates of the polymer blends were measured by UV-DSC (Mettler Toledo GmbH, Gießen, Germany) under N2 atmosphere. For each blend, 8–10 mg was tested in an aluminum pan without a lid at 25 °C. The heat flow was recorded over a time frame of 6 min (2 min before UV exposure, 3-min UV exposure, 1 min after exposure).
For scanning electron microscopy (SEM), the samples were sputter-coated with a conductive gold layer at 30 mA for 40 s using a Jeol JFC-1300 auto fine coater (Jeol Ltd., Akishima, Japan). Secondary electron images were acquired at 5 kV accelerating voltage in high vacuum conditions using an FEI Quanta 400 FEG (Thermo Fisher Scientific, Waltham, USA).

### Roll-to-roll processing

For continuous fabrication, a roll-to-roll system (Jakob Weiß & Söhne GmbH, Sinsheim, Germany) with a total web length of 24 m was used. The system consists of several units including coating, thermal drying, imprinting and UV curing and delamination. The imprinting unit is the core of the machine, which includes an imprinting roll and two backing rolls placed on each side; a pressure roll pressed the pre-polymer (i.e., wet coating) against the imprinting roll, and a deflection roll guided the delamination (Fig. 1a). The flexible transparent (tPDMS) or non-transparent (nPDMS) silicone templates were wrapped onto the imprinting roll. The UV-LED radiator with a maximum power of 6 W cm−2 at 365 nm was placed under the imprinting roll. 30% or 50% of UV power was used for different trials for comparison purposes. Detailed processing parameters for each experiment are summarized in Table 2.
Table 2
The parameters of roll-to-roll process
Samples
Ground layer
Process parameters
Top layer
Material
Material
UV power (%)
Thickness (µm)
Distancea (µm)
Mold material
90_(− 245 µm)
UA16
UA16
30/50
90
− 245
nPDMS
120_(− 245 µm), 120_(− 303 µm)
UA16
UA16
50
120
− 245, − 303
tPDMS, nPDMS
U6S9
U6L7
U6S9
50
120
− 245, − 303, − 362
nPDMS
U6E4
U6L7
U6E4
50
120
− 245, − 303, − 362
nPDMS
aRepresents distance between the flexible template and the ground layer. Positive value means that there is a gap between the flexible mold and ground layer; negative value means that the flexible mold is compressed
In the present work, the roll-to-roll manufacturing followed a two-step process as illustrated in Fig. 1b: First a non-patterned film (UA16 or U6L7) with a 30 µm wet coating thickness was formed onto the PET substrate (75 µm thickness, Melinex 506, DuPont Teijin Films Europe, Luxemburg) by covering the wet film with a release foil (polyethylene film, 23 µm Hostaphan MP 23, Mitsubishi Polyester Film GmbH, Wiesbaden, Germany) to protect the wet film against direct contact with the micro-patterning molds. After UV curing of this non-patterned ground layer, the release foil was removed. In a second step, the microstructures were imprinted based on coatings of UA16, U6S9 or U6E4 (90 or 120 µm wet coating thickness) as top coating on the ground layer. During imprinting, the distance between the ground layer and the flexible template, d, was adjusted by moving the pressure roll to control the filling of the microcavities by the pre-polymer. Positive values of d mean that there was a gap between the ground layer (without considering the top layer) and the flexible template, whereas negative values represent the flexible template being compressed. Upon UV curing, the microstructures were demolded from the flexible template.

Adhesion tests were carried out on a custom-built testing device as illustrated in Fig. 2a. The device is composed of an interferometer, a pivotable stage and a smooth glass sphere with curvature radius of 15 mm mounted on a glass beam with a spring constant of 2241 N m−1. Forces were calculated based on the beam deflection and the spring constant [24]. Samples with a size of 15 mm × 15 mm were cut from the micropatterned films and subsequently glued to a glass slide with a UV adhesive (BO MV76002, Bohle AG, Haan, Germany). A typical force–displacement curve for a normal pull-off experiment using a spherical probe is shown in Fig. 2b. The glass sphere was brought in contact with the samples at a velocity of 5 µm s−1 until a compressive preload of 30 mN was reached. The apparent contact area, A, was estimated from the radius of the glass sphere and the indentation depth (which is the difference between the displacement at preload and that achieved at first contact) from the equation: $$A = \pi \left[ {R^{2} - (R^{2} - \delta^{2} )} \right]$$, where R is the radius of the probe and $$\delta$$ is the indentation depth at preload. After preloading, the sphere was retracted in normal direction at a constant velocity of 5 µm s−1. The pull-off force was defined as the maximum tensile force before detachment. Pull-off stresses were determined by dividing pull-off forces by the apparent contact area, A, as described above. The work of separation was calculated by integrating the enclosed area of the stress–displacement curve in the tensile regime (see Fig. 2b). Six positions on each sample were measured, and the mean values were recorded.
Peel tests on smooth, flat glass substrates with a 90° peel angle were performed with 24-mm-wide and 240-mm-long films. The peeling speed was 5 mm s−1. For comparison with our tapes, magic tape (Scotch® Magic™ Tape, 3 M Deutschland GmbH, Neuss, Germany) was measured under the same condition. The normalized peel force was determined by dividing the measured forces by the width of the tapes.
Shear strength was examined by lap shear test in a tensile tester (Material Testing Machine Xforce P, Zwick & Roell GmbH & Co. KG, Ulm, Germany). The adhesive strip was attached on a glass plate with a contact area of 20 mm × 50 mm. The crosshead speed was 5 mm min−1. Shear strength was calculated by dividing the maximum shear force by the contact area.

## Results and discussion

### Process optimization

The two-step replication, as shown in Fig. 1b, was successfully implemented into the continuous roll-to-roll process: First, the plain ground layer (thickness 30 µm) was coated onto the PET carrier film. Second, the micropatterned adhesive layer was added. To study how the morphology as well as the adhesive properties of the composite film was affected by the thickness of the second layer, three different conditions were investigated: “90_(− 245 µm),” “120_(− 245 µm)” and “120_(− 303 µm).” The first number in the notation represents the layer thickness of the second wet layer (in µm), whereas the second number represents the set distance between the imprinting and the pressure roll during imprinting (see Fig. 1a). Negative distance implies compression between the template and the wet coating during the imprint. In this test, pure UA16 (without additives) was used for the ground and the second layer. The power of the UV curing lamp was set to 50%. Scanning electron micrographs of the microstructures obtained from the three different imprint conditions are presented in Fig. 3a. Pillar dimensions such as pillar height and diameter of the pillar top are shown in Fig. 3b. The figure includes three regimes, highlighting the pillar top diameter: “standard mushroom” with diameters between 67 and 72 µm, corresponding to the dimensions obtained from the Ni-shim master structure; “smaller mushroom” with diameters ranging between 56 and 67 µm; and “no mushroom” with diameters smaller than 56 µm associated with the absence of mushroom tips. Clearly, all three conditions tested in the first tests led to “small mushrooms” with diameters ranging from 57 µm (“90_(− 245 µm)” and “120_(− 245 µm)”) to 60 µm (“120_(− 303 µm)”). Thus, the diameter slightly increased with more negative distances, i.e., higher compressive loads between the imprinting and the pressure roll. The pillar heights ranged between 53 and 56 µm. The specimens made from 120-µm-thick wet coatings and the highest compressive loads ($$d = - 303 \upmu{\text{m}}$$) showed the best adhesion performance with a pull-off stress about 70 kPa and a work of separation about 1.25 J m−2 (Fig. 3c), which most probably is related to the largest mushroom caps obtained under these conditions [47].
In addition to compression during the imprint, UV-curing conditions are crucial for the cross-linking density and curing kinetics. We evaluated the impact of the UV power and the transparency of the silicone template (Fig. 4). Results for two different power ratings of the UV lamp are shown in (i) and (ii) in Figs. 4a–d. The micrographs reveal that microstructures exposed to 30% UV power exhibited no mushrooms (Fig. 4a). For 50% UV power, mushroom caps were obtained, but their size was still very small. Adhesion tests shown in Fig. 4c confirmed the results; by increasing the UV power to 50%, the pull-off stress was improved by 17% and the work of separation by 29%.
To further explore the effect related to UV exposure, the non-transparent silicone template (nPDMS) used in previous tests was replaced by a translucent silicone template (tPDMS). This modification led to much larger mushroom caps with tip diameters of up to 61 µm, as presented in (iii) and (iv) in Figs. 4a and d. In addition to larger caps, the structures were elongated with pillar heights of 59 µm, compared to 54 µm for structures generated by the non-transparent template. Figure 4e summarizes the adhesion performance of the micropatterned adhesives obtained using the different templates: Micropatterned adhesives replicated by using the translucent templates exhibited significantly higher pull-off stress (83 kPa) and work of separation (1.8 J m−2) compared to the adhesives generated by non-transparent templates with a pull-off stress of 70 kPa and a work of separation of 1.3 J m−2. The larger mushroom caps accompanied with the enhanced adhesion performance are likely related to better penetration of UV light through the translucent template. Thus, the complete mushroom-shaped microstructure including the undercut of the mushroom cap was homogeneously cured. In addition, the transparent mold was less flexible (approx. two times stiffer, see Table 1) compared to the non-transparent template. Thus, the tPDMS template is less compressed during the imprint, which most probably explains the longer pillars. As a disadvantage, we observed that, after several imprint cycles, microstructures ruptured during demolding from the translucent template (tPDMS) and stuck inside the cavities. An explanation for this is that UA16 resin diffused into the tPDMS template upon repeated imprints. This can result in strong covalent bonds between the pre-polymer inside the cavity and the interpenetrated reactive oligomers, which in turn led to cohesive rupture of the microstructures. In contrast, repeated imprinting using the nPDMS was always successful without any residues inside the cavities upon demolding.

### Material optimization

Besides process parameters, different material compositions (UA16-based polymer blends) were evaluated to further improve the formation of mushroom caps by enhancing the UV-curing kinetics. The parameters of the replication process for the different blends are summarized in Table 2. The kinetics of the UV-induced curing was modified by adding to UA16 either 1% Miramer SIP900 (polymer blend: U6S9) or 20% Miramer ES4420 (polymer blend: U6E4). Figure 5a shows the results of the photo-differential scanning calorimetry (UV-DSC), which implies that U6E4 cross-linked more quickly compared to U6S9 and UA16. In addition, mercapto-modified acrylate oligomers (U6E4) are well known to reduce the oxygen inhibition during a radical polymerization reaction [48]. In contrast to the tests presented above, the composition of the ground (first) layer was modified by adding 10% Laromer 8887 (polymer blend: U6L7), i.e., a mono-functional acrylic acid ester. Laromer 8887 acted as “thinner” [30, 33] by reducing the cross-linking density accompanied with polymer stiffness reduction and lower elongation at break (see values in Table 1). Micrographs of microstructures obtained from U6S9 (i–iii) and U6E4 (iv–vi) in terms of varying pressing distances are shown in Fig. 5b. Dimensions of the microstructures are reported in Fig. 5c. For U6S9, the diameter of the pillar diameter increased from 58 to 66 µm with decreasing pressing distance from − 245 to − 363 µm. For U6E4, the mushroom caps were larger and less sensitive to the pressing distances. Thus, diameters of the mushroom tips varied between 65 and 67 µm. The adhesion experiments confirmed these trends: The pull-off stresses obtained for U6S9 increased from 56 to 87 kPa with decreasing pressing distance from − 245 to − 363 µm. The work of separation was about 2 ± 0.2 J m−2 for all U6S9 specimens. The best adhesion performance was obtained for U6E4 with the highest pull-off stress of 116 kPa and the highest work of separation of 3.78 J m−2 for specimens made with a pressing distance of − 303 µm.

### Application demonstrations

Following process and material optimization, micropatterned adhesive films of U6E4 were continuously fabricated at a pressing distance of −303 µm. The adhesion properties and durability of the adhesive films obtained were further characterized. Figure 6 shows the peel adhesion of the micropatterned U6E4 film in comparison with 3 M Scotch Magic tape. For both films, we found similar peel forces between 1 and 1.4 N cm−1 for a 90° peel angle. Inspection upon tests revealed residue-free detachment for both films. Using lap shear test, a shear strength of the micropatterned U6E4 film of about 71 ± 9 kPa was obtained. In Fig. 7a, the reusability and shear adhesion of a 20-mm-wide U6E4 film were demonstrated by handling a glass object (1.163 kg): Upon attachment of the film, the object was lifted (shear-loaded), placed at the target position and the adhesive film was peeled off and attached again. The procedure was successfully repeated 5 times without any damage of the structure nor residues at the glass surface. Figure 7b demonstrates a similar test for the normal loading direction. Here, upon attachment, the glass object was normally lifted and placed at the target position. The cycle including attachment, lifting and detachment was successfully repeated 5 times without any visible damages nor residues.
The durability of the micropatterned dry adhesive was examined by a long-term pick-and-place test. A circular pad of adhesive film (U6E4, produced with d = − 303 µm; diameter 30 mm) repeatedly picked up and released a glass plate with mass of 145 g for 100,000 attachment–detachment cycles (see Fig. 8a). After these cycles, the micropatterned structure exhibited a certain extent of abrasion and contamination (see Fig. 8b). Similarly, the pull-off stress decreased from nearly 120 kPa to around 90 kPa (see Fig. 8c). However, the pick-and-place action can be still well performed even after 100,000 cycles, demonstrating a good endurance of PUA dry adhesive.

## Conclusions

The present paper has shown that achieving optimum mushroom tips can be a challenge for fabrication by roll-to-roll processing. The results presented demonstrate that the formation of mushroom caps during imprinting is strongly connected with the UV-curing conditions and the materials used. The following conclusions can be drawn:
• Process parameters for optimum mushroom tips: The absence of mushroom tips is related to insufficient UV curing, but the quality of the replicated microstructures can be improved by increasing UV power, the choice of translucent templates or adding of chemical components accelerating the curing kinetics. For example, cap size was much larger by using the translucent template, but microstructures stuck inside the cavities after several imprint cycles, being inacceptable for continuous fabrication.
• Effect of UV treatment on adhesion performance: Only mushroom tips that were completely replicated in their size show high adhesion performance. By increasing UV power from 30 to 50%, the pull-stress was improved by 17% and work of separation by 29%. Replacing opaque template with translucent template resulted in a pull-off stress increase by 19%.
• Effect of material modification on adhesion performance: Adding polyester-modified silicone acrylates (Miramer SIP900) or mercapto-modified acrylates (Miramer ES4420) accelerated the cross-linking kinetics. In particular, the polymer blend U6E4 provided best results in mushroom cap sizes similar to the master structure and excellent adhesion performance: pull-off stress of 116 kPa work of separation of 3.78 J m−2, significantly larger compared to structures made from pure, unmodified UA16.
• Comparison of adhesion performance: A peel strength of 1.4 N cm−1 similar to commercial Scotch Magic tape and a shear strength of 71 kPa were achieved with the optimized micropatterned adhesive. Repeated attachment–detachment cycles without loss of the functionality proved reliable usability and robustness of the micropatterned dry adhesive for versatile application.

## Acknowledgements

This work was funded by the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Advanced Grant Agreement no. 340929 and by the Leibniz-Competition Grant no. 493. We thank Jennifer Dollmann for the dynamic mechanical thermal analysis, Marcus Koch for the help on SEM measurements, Florent Zogaj and Sebastian Meyer for their assistance on adhesion measurements.

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### Conflict of interest

The authors declare that they have no conflict of interest.

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Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

• über 69.000 Bücher
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aus folgenden Fachgebieten:

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Sie erhalten uneingeschränkten Vollzugriff auf alle acht Fachgebiete von Springer Professional und damit auf über 45.000 Fachbücher und ca. 300 Fachzeitschriften.

### Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

• über 50.000 Bücher
• über 380 Zeitschriften

aus folgenden Fachgebieten:

• Automobil + Motoren
• Bauwesen + Immobilien
• Business IT + Informatik
• Elektrotechnik + Elektronik
• Energie + Umwelt
• Maschinenbau + Werkstoffe

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