Slide 1

Phase transitions

If you heat a pot of water on the stove ‘till it boils, what do you see? More to the point, what can you see through? The liquid water is clear, and the steam is clear. And you can tell the two apart.
PICTURE OF WATER BOILING ON STOVE.
Liquid water and steam (gaseous water) are two phases of the same substance. To boil water is to cause a phase transition because we start with one phase (liquid) and end up with another. There’s another special thing about this phase transition – it’s not hard to tell the liquid and gaseous phases apart, and there is both liquid and gaseous water in the pot at the same time. If we went in the opposite direction – cooling the steam to liquid – we’d see droplets of water condense bit by bit, just like we see bubbles of steam appear. Either way, the transition doesn't happen everywhere all at once. It happens in bits and pieces, and takes some time and a certain amount of energy (in the form of heat) before all the water has turned to gas.
PHASE DIAGRAM (PT and PV)‏
Look at the phase diagram above; it shows you what temperatures and pressures water exists as a solid (ice), as a liquid, and as a gas (steam). A red line has been drawn to show the boiling transition on the phase diagram.
Ask yourself:
- What are some phase transitions you see all the time? What do they have in common?
- Why do we see the bubbles in the boiling water if liquid and gaseous water are both see-through?
- Click here for the answers!
Quizlet:
- Is an ice cube melting in a drink a phase transition?
- Is blending strawberries, bananas, and milk to make a smoothie a phase transition?
- Is burning a piece of wood a phase transition?
- Click here for the answers!

Phase transitions

If you heat a pot of water on the stove ‘till it boils, what do you see? More to the point, what can you see through? The liquid water is clear, and the steam is clear. And you can tell the two apart.
PICTURE OF WATER BOILING ON STOVE.
Liquid water and steam (gaseous water) are two phases of the same substance. To boil water is to cause a phase transition because we start with one phase (liquid) and end up with another. There’s another special thing about this phase transition – it’s not hard to tell the liquid and gaseous phases apart, and there is both liquid and gaseous water in the pot at the same time. If we went in the opposite direction – cooling the steam to liquid – we’d see droplets of water condense bit by bit, just like we see bubbles of steam appear. Either way, the transition doesn't happen everywhere all at once. It happens in bits and pieces, and takes some time and a certain amount of energy (in the form of heat) before all the water has turned to gas.
PHASE DIAGRAM (PT and PV)‏
Look at the phase diagram above; it shows you what temperatures and pressures water exists as a solid (ice), as a liquid, and as a gas (steam). A red line has been drawn to show the boiling transition on the phase diagram.
Ask yourself:
- What are some phase transitions you see all the time? What do they have in common?
- - Here are a few: rain, snow, seeing your breath on a cold day, turning on a fluorescent light or a plasma TV, dissolving sugar into tea... You might even think about car traffic as having different phases – standstill, slow, fast...
- Why do we see the bubbles in the boiling water if liquid and gaseous water are both see-through?
- - Molecules in liquid water are much closer to each other than in gaseous water. So the density of liquid water is greater than that of gaseous water. Very often, light travels slower through denser media than through less dense media. At the boundary between “slow” (dense) and “fast” (less dense) media, light bends – this is called refraction. This is what causes haze or mirages on a hot day, and what makes glasses and telescopes work. And, except in a few special circumstances, not all of the light goes through the boundary – a part of it (sometimes all of it) bounces back, or is reflected. This is why you can see your reflection in a window while still seeing through the window. And because some light bounces off the steam bubbles in the liquid water and into our eyes, we can see the bubbles in boiling water.
Quizlet:
- Is an ice cube melting in a drink a phase transition?
- Is blending strawberries, bananas, and milk to make a smoothie a phase transition?
- Is burning a piece of wood a phase transition?
- Click here for the answers!

Phase transitions

If you heat a pot of water on the stove ‘till it boils, what do you see? More to the point, what can you see through? The liquid water is clear, and the steam is clear. And you can tell the two apart.
PICTURE OF WATER BOILING ON STOVE.
Liquid water and steam (gaseous water) are two phases of the same substance. Boiling water is a phase transition because we start with one phase (liquid) and end up with another. There’s another special thing about this phase transition – it’s not hard to tell the liquid and gaseous phases apart, and there is both liquid and gaseous water in the pot at the same time. If we went in the opposite direction – cooling the steam to liquid – we’d see droplets of water condense bit by bit, just like we see bubbles of steam appear. Either way, the transition doesn't happen everywhere all at once. It takes some time and a certain amount of energy (in the form of heat) before all the water has turned to gas.
PHASE DIAGRAM (PT and PV)‏
Look at the phase diagram above; it shows you what temperatures and pressures water exists as a solid (ice), as a liquid, and as a gas (steam). A red line has been drawn to show the boiling transition on the phase diagram.
Ask yourself:
- What are some phase transitions you see all the time? What do they have in common?
- Why do we see the bubbles in the boiling water if liquid and gaseous water are both see-through?
- Click here for the answers!
Quizlet:
- Is an ice cube melting in a drink a phase transition?
- - Yes! The solid water changes into liquid.
- Is blending strawberries, bananas, and milk to make a smoothie a phase transition?
- - Yes and no – chopping up the strawberries or bananas into really small pieces would not be a phase transition, just like cutting one piece of wood into two is not a phase transition. But dissolving the sugars from these fruits into the milk is a phase transition.
- Is burning a piece of wood a phase transition?
- - No! When wood is burned, it is transformed into a different substance – the molecules that make up the wood actually change into different molecules (i.e. from cellulose to carbon monoxide, carbon dioxide, and water). This is an example of a chemical reaction. A phase transformation occurs when one substance turns into a different form of itself. Every water molecule in pure ice, liquid water, and steam is chemically the same – each one is made up of one oxygen atom and two hydrogen atoms. What's different in these phases is the way the molecules are arranged.

Critical Point

DUDE!
- Now here’s the weird part – at high temperatures and pressures, it becomes harder to tell liquid water and gaseous water apart. There’s a special mix of temperature and pressure – we call it the critical point – where the difference between liquid and gas ceases to exist. For water, this happens at 374 degrees Celsius (705 degrees Fahrenheit) and 218 atmospheres (normal air pressure is one atmosphere at sea level!). For carbon dioxide, it’s 31.1 °C and 73 atmospheres. For nitrogen, -147 °C and 33 atmospheres. No wonder it seems so funky to us! For a bunch of the substances in our world, the critical point happens at conditions extreme for human beings. We’d never see it in the kitchen!
PHASE DIAGRAM
- In the phase diagram, the area where liquid and gas combine into the same thing is labeled as the “supercritical” region – in this case, “super” means “above”.
Quizlet:
- What was the important difference between liquid and gas in the first place?
- - (a) One is wet and the other dry.
  - (b) One is cold and the other hot.
  - (c) One is more dense than the other.
  - (d) There's no difference!
- Click here for the answer!
PICTURE of density.
The word, density, keeps coming up. Remember what it means? It's how much matter (stuff) there is in a certain volume (space). For this liquid-gas transition, density plays the role of the order parameter - the useful property which sets the two phases apart. Different situations call for different order parameters, and we’ll see another order parameter when we think about phase transitions in magnets.
At the critical point, the density of liquid and gas become the same. And so the difference between these two phases disappears – they become a single, supercritical phase.

Critical Point

DUDE!
- Now here’s the weird part – at high temperatures and pressures, it becomes harder to tell liquid water and gaseous water apart. There’s a special mix of temperature and pressure – we call it the critical point – where the difference between liquid and gas ceases to exist. For water, this happens at 374 degrees Celsius (705 degrees Fahrenheit) and 218 atmospheres (normal air pressure is one atmosphere at sea level!). For carbon dioxide, it’s 31.1 °C and 73 atmospheres. For nitrogen, -147 °C and 33 atmospheres. No wonder it seems so funky to us! For a bunch of the substances in our world, the critical point happens at conditions extreme for human beings. We’d never see it in the kitchen!
PHASE DIAGRAM
- In the phase diagram, the area where liquid and gas combine into the same thing is labeled as the “supercritical” region – in this case, “super” means “above”.
Quizlet:
- What was the important difference between liquid and gas in the first place?
- - (c) One is more dense than the other. – Molecules in a liquid are spaced closer together than in a gas, so the density of a liquid is greater than that of a gas. There are two processes at work here: one is the natural attraction that molecules feel toward one another (yup!), and the other thermal energy – random motion of the molecules. When thermal motion is more important (at higher temperatures), molecules bounce all over the place rather than sticking together, and so they take up more space. At lower temperatures, the attraction between molecules is more important, and so molecules stay close together.
PICTURE of density.
The word, density, keeps coming up. Remember what it means? It's how much matter (stuff) there is in a certain volume (space). For this liquid-gas transition, density plays the role of the order parameter - the useful quantity which sets the two phases apart. Different situations call for different order parameters, and we’ll see another order parameter when we think about phase transitions in magnets.
At the critical point, the density of liquid and gas become the same. And so the difference between these two phases disappears – they become a single, supercritical phase.

Clues

<Picture of sleuth with magnifying glass, Sherlock Holmes, etc.>
Critical opalescence is a terrific clue to what’s happening at the critical point because it tells us that something is scattering visible light. Light goes into the substance, but instead of going through like normal, it bounces off into all different directions. If a clear normally substance is scattering light, something must be going on inside the substance, but what?
To figure out what, we need to understand why light scatters in the first place. Important discoveries about the scattering of light were made by John Tyndall in the 1860s, and by John Strutt (Lord Rayleigh), in 1871. The thing we need to know about scattering right now is that small changes or imperfections in an otherwise uniform, clear medium scatter light. Here, we know the medium – be it water or carbon dioxide or whatever system we’re looking at – is usually uniform, but must develop lots and lots of changes inside close to the critical point.
REMEMBER...
- Remember why we were able to see the bubbles of steam in boiling liquid water? Keep this in mind – it'll come in handy....
Another important clue comes from the phase diagram.
PHASE DIAGRAM OF WATER (PV)
Look at the phase diagram. It is drawn with pressure on the vertical axis and volume on the horizontal axis. The lines you see show what pressure you get for a given volume at a certain temperature. They are called isotherms because each line tells you all of the pressures and volumes you get at a certain temperature (from the Greek prefix, iso, meaning same, and word, thermos, meaning heat).
The critical isotherm is just the isotherm drawn at the critical temperature. This isotherm is special because there is a certain temperature and pressure for which it becomes flat. Here, “flatness” is measured by the slope (rise over run) – the flatter the line is, the smaller the slope. None of the isotherms with temperature greater than critical are flat.
The inverse of the slope (one divided by the slope) – run over rise – tells you how far you move along the horizontal axis if you change your spot on the vertical axis. But remember, these axes have meanings – the horizontal axis is volume, and the vertical is pressure. So the inverse of the slope tells you the change in volume for a little change in pressure. This ratio has a name – the isothermal compressibility (or how “press-able” something is at a certain temperature). Something with a small compressibility requires a lot of pressure to change its size – it is very stiff. Something with a large compressibility changes size very readily – it is very soft.
PICTURE: Person squeezing a rock, person squeezing a pillow. Labelled, respectively, low compressibility and high compressibility.
If the isotherm is very flat, and the slope very small, then the inverse of the slope is very big. In fact, on the critical isotherm, the compressibility becomes infinitely large (at least, that’s what this simple model tells us). That means that even a tiny change in pressure results in a very large change in volume. If the volume changes, but the mass stays the same, then the density changes.
The key to understanding critical opalescence is to recognize that there are always small changes, or fluctuations, in quantities like pressure or temperature. This is because the liquid or gas that we’re studying is actually composed of lots and lots and lots (and lots) of little, tiny elements.

The BIG Deal

How big are taling about? How many molecules?
- PICTURE of hourglass)
- Well, a lot. – think in terms of 6 * 1023 – that’s Avogadro’s number, or about how many hydrogen nuclei there are in one gram. That number is a six followed by 23 zeros, or.
- It’s hard to understand just how many this is. But to give you an idea, there are about six billion, or 6 * 109, people on the planet. There are about 150*109 meters between Earth and the sun. In the Milky Way Galaxy, there are fewer than one trillion stars (1012). And the gross domestic product of the world was about $5.4*1013 in 2007. All of these numbers are not anywhere close to Avogadro’s number. If you were to count every grain of sand on the planet, you might start to get close (but by the time you finished counting, there’d be even more sand, and you’d have to start all over!).
- TABLE:
- # hydrogen nuclei in one gram: 600 000 000 000 000 000 000 000
- # $ changing hands in the world 2007: 54 000 000 000 000
- Distance from Earth to sun (meters): 150 000 000 000
- Number of humans on Earth: 6 000 000 000
- The point is that these quantities – temperature, pressure, compressibility – are kind of taking an average of the behavior of this very large system.
- You can find the average height of people in your classroom. If there are only two or three people in the class, this average height doesn’t mean a lot because there is probably a lot of variation about the average. On the other hand, if you have a really big class, say fifty, the average starts to be a more useful quantity. But there are always variations about the average – some people who are a little taller or shorter than average.
- (PICTURE of cartoon of people with different heights?)‏
- Same deal with pressure and temperature. For example, in a gas, particles are always whizzing around and bumping into each other, into walls, and so forth. Temperature tells us something about how fast the average particle is travelling at any given time. Actually, not all the particles are travelling the average velocity, just like not everyone in your class is of average height. But the average is still a useful number.
- It turns out that, under normal circumstances, the variation about the average gets smaller as the number of molecules gets bigger. So if the number of molecules is really big (like 6*10^23), the variation is really small.
- But what if there were something that would be totally changed by even tiny fluctuations? Say, for example, you are cooking some really hard-to-make dish, like sugar candy. A small change in temperature can have a really big effect on what the finished product looks like.
- What’s the analogy for critical phenomena? The response to a little variation from the average is huge in at the critical point. We already saw this from the compressibility – the response of volume to a small change in pressure. So little variations about the average become really big.
What does this have to do with critical opalescence?
- Remember that little changes inside an otherwise uniform medium can scatter light? And remember that we were able to see steam bubbles in boiling liquid water because these two phases have different densities, and light bounces off the boundary between bubble and liquid?
- That’s what happens at critical opalescence. There are always lots of very small variations (fluctuations) about the average quantities like temperature, pressure, and so on. These are usually too small to detect – too small to scatter light. But around the critical point, response functions like compressibility get really huge – this acts like a magnifying glass for fluctuations. So right around the critical point, really big fluctuations in density appear – big enough to scatter light.
- It was Albert Einstein and Marian Smoluchowski who made this connection around 1910.
Why’s this important? It’s just a dirty-looking gassy-liquid thing?
- For a couple of reasons. First, around the time when Smoluchowski and Einstein were working on this problem, not everyone was convinced there existed such a thing as a molecule. After all, you can’t see a single air or water molecule with the human eye, can’t feel it with your hands. Materials usually don’t look like they come in little bits and pieces – they look like they fill up a space completely – that they’re continuous.
- But phenomena like critical opalescence make a lot more sense in the molecular picture of the world than the continuous one. And so the study of critical opalescence help to explain something very fundamental about the nature of matter.
- It’s hard to appreciate this in the twenty-first century because we hear words like “atom” and “molecule” all the time, But this was a great (and very, very old!) debate in the early twentieth century – one of the reasons why “Einstein” is a household name is that Einstein helped to resolve the question.
- Critical opalescence is also important because it’s part of something much, much larger. It’s called “universality”, and we’ll get back to it later.
Is this the end of the story?
- Nope – the beginning. Fortunately, in science, every time you answer a question, you get a bunch more!
Quizlet:
- Under “normal” circumstances, temperature and pressure become more “exact” representations of a bunch of particles as
- - (a) the sensitivity of the thermometer or barometer is improved.
  - (b) the system approaches the critical point.
  - (c) the number of particles gets smaller.
  - (d) the number of particles gets bigger.
- Critical opalescence
- - (a) occurs for first-order phase transitions
  - (b) occurs when a fluctuation gets repeated a long distance away, so that fluctuations are big enough to scatter light
  - (c) is an essential reduction of opal gems (essence of opals)‏
- Light scatters at the critical temperature because
- - (a) Density variations become very large, and light travels through different densities at different speeds.
  - (b) The liquid water starts to boil or the gas to condense, and the steam bubbles or droplets are bigger than usual.
  - (c) The light photons hit each other and bounce everywhere.
  - (d) Einstein says so.
External links:
- http://www.ucl.ac.uk/~uccaata/work/opalescence/opalescence.html
- http://www.msm.cam.ac.uk/doitpoms/tlplib/solid-solutions/demo.php
- http://en.wikipedia.org/wiki/Critical_opalescence
- http://www.chem1.com/chemed/critical.html
- http://en.wikipedia.org/wiki/Rayleigh_Scattering

The BIG Deal

How big are taling about? How many molecules?
- PICTURE of hourglass)
- Well, a lot. – think in terms of 6 * 1023 – that’s Avogadro’s number, or about how many hydrogen nuclei there are in one gram. That number is a six followed by 23 zeros, or.
- It’s hard to understand just how many this is. But to give you an idea, there are about six billion, or 6 * 109, people on the planet. There are about 150*109 meters between Earth and the sun. In the Milky Way Galaxy, there are fewer than one trillion stars (1012). And the gross domestic product of the world was about $5.4*1013 in 2007. All of these numbers are not anywhere close to Avogadro’s number. If you were to count every grain of sand on the planet, you might start to get close (but by the time you finished counting, there’d be even more sand, and you’d have to start all over!).
- TABLE:
- # hydrogen nuclei in one gram: 600 000 000 000 000 000 000 000
- # $ changing hands in the world 2007: 54 000 000 000 000
- Distance from Earth to sun (meters): 150 000 000 000
- Number of humans on Earth: 6 000 000 000
- The point is that these quantities – temperature, pressure, compressibility – are kind of taking an average of the behavior of this very large system.
- You can find the average height of people in your classroom. If there are only two or three people in the class, this average height doesn’t mean a lot because there is probably a lot of variation about the average. On the other hand, if you have a really big class, say fifty, the average starts to be a more useful quantity. But there are always variations about the average – some people who are a little taller or shorter than average.
- (PICTURE of cartoon of people with different heights?)‏
- Same deal with pressure and temperature. For example, in a gas, particles are always whizzing around and bumping into each other, into walls, and so forth. Temperature tells us something about how fast the average particle is travelling at any given time. Actually, not all the particles are travelling the average velocity, just like not everyone in your class is of average height. But the average is still a useful number.
- It turns out that, under normal circumstances, the variation about the average gets smaller as the number of molecules gets bigger. So if the number of molecules is really big (like 6*10^23), the variation is really small.
- But what if there were something that would be totally changed by even tiny fluctuations? Say, for example, you are cooking some really hard-to-make dish, like sugar candy. A small change in temperature can have a really big effect on what the finished product looks like.
- What’s the analogy for critical phenomena? The response to a little variation from the average is huge in at the critical point. We already saw this from the compressibility – the response of volume to a small change in pressure. So little variations about the average become really big.
What does this have to do with critical opalescence?
- Remember that little changes inside an otherwise uniform medium can scatter light? And remember that we were able to see steam bubbles in boiling liquid water because these two phases have different densities, and light bounces off the boundary between bubble and liquid?
- That’s what happens at critical opalescence. There are always lots of very small variations (fluctuations) about the average quantities like temperature, pressure, and so on. These are usually too small to detect – too small to scatter light. But around the critical point, response functions like compressibility get really huge – this acts like a magnifying glass for fluctuations. So right around the critical point, really big fluctuations in density appear – big enough to scatter light.
- It was Albert Einstein and Marian Smoluchowski who made this connection around 1910.
Why’s this important? It’s just a dirty-looking gassy-liquid thing?
- For a couple of reasons. First, around the time when Smoluchowski and Einstein were working on this problem, not everyone was convinced there existed such a thing as a molecule. After all, you can’t see a single air or water molecule with the human eye, can’t feel it with your hands. Materials usually don’t look like they come in little bits and pieces – they look like they fill up a space completely – that they’re continuous.
- But phenomena like critical opalescence make a lot more sense in the molecular picture of the world than the continuous one. And so the study of critical opalescence help to explain something very fundamental about the nature of matter.
- It’s hard to appreciate this in the twenty-first century because we hear words like “atom” and “molecule” all the time, But this was a great (and very, very old!) debate in the early twentieth century – one of the reasons why “Einstein” is a household name is that Einstein helped to resolve the question.
- Critical opalescence is also important because it’s part of something much, much larger. It’s called “universality”, and we’ll get back to it later.
Is this the end of the story?
- Nope – the beginning. Fortunately, in science, every time you answer a question, you get a bunch more!
Quizlet:
- Under “normal” circumstances, temperature and pressure become more “exact” representations of a bunch of particles as
- - (d) the number of particles gets bigger. – In fact, the variance scales as 1/N, where N is the number of particles. Now 1/10 = 0.1, 1/100=0.01, and 1/1*10^23 = 0.00000000000000000000001.
- Critical opalescence
- - (b) occurs when a fluctuation gets repeated a long distance away, so that fluctuations are big enough to scatter light. – This is tricky. The idea is that the density in one spot is connected with the density far away – the two are correlated. Think of a movie theater. Two people sitting next to each other might be having a conversation, whispering to each other. Two people sitting across the room might also be whispering and having a totally different conversation. But if they're talking right after the movie ended, chances are they're talking about the same thing – the movie. Their conversations are then correlated, and this same conversation appears again and again, all over the theater – sort of like a magnifying glass for this conversation. The analogy with the critical point is that all the molecules are doing their own thing 'till close to criticality – then, disturbances get repeated over long distances – they are correlated over large lengths. These disturbances are responsible for scattering light, giving rise to critical opalescence.
- Light scatters at the critical temperature because
- - (a) Density variations become very large, and light travels through different densities at different speeds. – Sudden changes in density can cause light to bounce – we talked about this when thinking about why we could see steam bubbles in water. The trick here is that the density differences are not binary – that is, there aren't two unique phases, and pockets or islands of one appear in the other. Instead, the variations which are always present in any collection of molecules, but are usually too small to matter, get magnified to the point where they scatter light. This is evidence that things like air and water are actually discrete – made up of lots of tiny particles – and not continuous blobs of matter.
External links:
- http://www.ucl.ac.uk/~uccaata/work/opalescence/opalescence.html
- http://www.msm.cam.ac.uk/doitpoms/tlplib/solid-solutions/demo.php
- http://en.wikipedia.org/wiki/Critical_opalescence
- http://www.chem1.com/chemed/critical.html
- http://en.wikipedia.org/wiki/Rayleigh_Scattering

Try it!

The following experiment is described in a paper by E.S.R. Gopal of the Indian Institute of Science which you can download for free (Resonance 5 (4), April 2000, 37-45).
We said that the critical point of many common substances is beyond the temperatures and pressures of the human habitat. BUT that doesn’t mean that there aren’t systems with critical points at comfortable temperatures and pressures. One such system is the binary liquid (“two liquid”) combination of methyl alcohol, CH3OH, and carbon disulfide, CS2. Where before, we thought about the liquid, gas, and supercritical phases of the water example, now, we think in terms of two substances either being dissolved in one another or separate. Above the critical temperature, methyl alcohol dissolves in carbon disulfide – one single phase. Below it, the two substance separate into two distinct liquids – two phases. The same kind of opalescence happens at the critical point for the same reason.
Experiment:
- In a glass beaker, combine methyl alcohol, CH3OH, with carbon disulfide, CS2, in the weight ratio of 20:80. Seal the beaker with a stopper.
- Heat the mixture in a bath of warm water to about 60 degrees Celsius – well above the critical point. The methyl alcohol should dissolve in the carbon disulfide so that the mixture appears as one liquid phase.
- Slowly cool the mixture down. Close to the critical point at 36 degrees Celsius, you should start to see a cloudy area appear. This is where the curved boundary between the two substances – a meniscus – will eventually form.
- - The cloud will get bigger and bigger until, right at the critical point, it takes up most of the space of the mixture.
  - Once the temperature crosses below the critical point, two distinct regions in the mixture should begin to appear, with a dark region in the cloud separating them.
  - Finally, well below the critical point, there should be two clearly distinguishable liquids, one sitting on top of the other.
  - Don’t worry if the critical point isn’t exactly at 36 degrees Celsius! It’s location is very sensitive to impurities in the CH3OH and CS2 – even a very little bit of water in the mix can change the critical temperature by a couple of degrees Celsius.

Universality

When we started talking about critical opalescence, we spoke of water and carbon dioxide and the gas-liquid transition. The experiment described something different – two different liquids going through a “dissolved/not-dissolved” transition. But everything we understood about critical phenomena from the liquid-gas transition seemed to work in the experiment.
The connection is deeper than that. If you look at the way response functions like compressibility get big around the critical point (how fast as you bring the temperature close to the critical point), you find that you end up with the same numbers for lots of very diverse situations.
As scientists, we have to take one more important step: we have to be a little more concrete in the description of “sameness”. This requires a little math, but it’s totally worth the effort. So here it goes: things get big (diverge) around the critical temperature as
PICTURE: EqCritExp.
Ask yourself...
- What happens to this quantity as T gets closer to Tc?
- There are three lines below corresponding to three different values of the critical exponent, x. Label the axes and match the line to the critical exponent.
PICTURE: Plots of EqCritExp. Without Labels.
The remarkable thing about critical phenomena is that a bunch of different kinds of quantities – response functions like comrpessibility and others – follow this rule close to the critical temperature. There might be a different critical exponent for each different kind of quantity, but the same kind of expression represents the behavior of all of these quantities.
The even more remarkable thing is that the critical exponents – the x's – for totally different systems are the same.
PICTURE of some face expressing “WOW”
You might not expect carbon dioxide and water to have the same set of critical exponents, because they’re two different substances. But they do, and, in fact, these same numbers would come out of the binary liquid experiment. There are more exotic-sounding phase transitions, too, like those of binary metal alloys or the paramagnetic-ferromagnetic crossover. These things seem really different from one another (think how different water is from methyl alcohol – you need to drink water or you’ll die, and you die if you drink methyl alcohol! And what did that experiment have to do with magnets?), yet the numbers coming out of experiments are exactly the same.
Ask yourself…
- What happens at the critical point that is always the same?
- Click here for the answer!
To gain further understanding, we need to open up this black box of fluctuations to figure out what is happening around the critical point that makes them grow so large.

Universality

When we started talking about critical opalescence, we spoke of water and carbon dioxide and the gas-liquid transition. The experiment described something different – two different liquids going through a “dissolved/not-dissolved” transition. But everything we understood about critical phenomena from the liquid-gas transition seemed to work in the experiment.
The connection is deeper than that. If you look at the way response functions like compressibility get big around the critical point (how fast as you bring the temperature close to the critical point), you find that you end up with the same numbers for lots of very diverse situations.
As scientists, we have to take one more important step: we have to be a little more concrete in the description of “sameness”. This requires a little math, but it’s totally worth the effort. So here it goes: things get big (diverge) around the critical temperature as
PICTURE: EqCritExp.
Ask yourself...
- What happens to this quantity as T gets closer to Tc?
- - T gets close to Tc, the function diverges or “blows up” (gets really big)
- There are three lines below corresponding to three different values of the critical exponent, x. Match the line to the critical exponent.
- - The critical exponent controls how fast the function blows up. The bigger the exponent, the faster the blows up. A gas-liquid critical point, like that for water or carbon dioxide, is associated with a bunch of different critical exponents – one for isothermal compressibility, one for heat capacity, and so on. We just recycle the same function with a different x.
PICTURE: Plots of EqCritExp. With Labels.
The remarkable thing about critical phenomena is that a bunch of different kinds of quantities – response functions like compressibility and others – follow this rule close to the critical temperature. There might be a different critical exponent for each different kind of quantity, but the same kind of expression represents the behavior of all of these quantities.
The even more remarkable thing is that the critical exponents – the x's – for totally different systems are the same.
PICTURE of some face expressing “WOW”
You might not expect carbon dioxide and water to have the same set of critical exponents, because they’re two different substances. But they do, and, in fact, these same numbers would come out of the binary liquid experiment. There are more exotic-sounding phase transitions, too, like those of binary metal alloys or the paramagnetic-ferromagnetic crossover. These things seem really different from one another (think how different water is from methyl alcohol – you need to drink water or you’ll die, and you die if you drink methyl alcohol! And what did that experiment have to do with magnets?), yet the numbers coming out of experiments are exactly the same.
Ask yourself…
- What happens at the critical point that is always the same?
- - What all of these situations have in common are fluctuations about an average state (temperature and pressure). Right around the critical point, these fluctuations become magnified, and control the way the system behaves. Because the fluctuations grow in the same way – whether they are fluctuations of average density or magnetization or any other order parameter – the same rate of growth appears for all of these systems….WOW!
  - ….or something like that – it’s a little more complicated. It turns out that the critical behavior is not always the same, but it doesn’t depend so much on the identity of the substance at play – more on its dimensionality and number of components.
  - WHAT?
  - More on this later.
To gain further understanding, we need to open up this black box of fluctuations to figure out what is happening around the critical point that makes them grow so large.

It’s everywhere!

Fractals are ubiquitous in nature and society. Think of it: a big object which is made up of many small objects similar to the big object - sound like anything? (Say the United Nations, made up of countries like the United States, made up states like New York, made up of counties like New York, made up of cities like New York, made up of boroughs, made up New Yorkers. The organizational body of the UN needs to secure resources and quality of life for its members, as does each New Yorker for his or her family, etc....)
The lesson to learn is that scale invariance - fractality - is not an esoteric concept filling the rarified space of the academy's upper echelons, nor is it some exotic thing found only in the extreme situations - it's just a reasonable idea that appears often in the world. Recognizing it helps us understand the nature of things. And knowing how to understand it inspires us to think of some clever ways to solve problems!
Another lesson to learn is that even things which seem to be very different may, in fact, share common features at the most basic level. In other words, science is a fractal! So is society!
Fractals in nature...
second-order phase transitions (critical opalescence, ferromagnetic/paramagnetic phase transition, etc.)
Properties of subatomic particles (particle physics)
biology (ferns, DNA, feet of those lizard/amphibians which stick to walls, etc. ...)
Snowflakes, lightning,
Mountains, coastlines
cracks
where it happens in technology and society...
-To clarify, we mean where the concept of scale invariance is applied directly to some description of a manmade object or to society; certainly, physics still permeates....
Using framework to understand:
/economics, finance
/In networks: Internet traffic,
/Industrial processing
Architecture, music, art
/Cracks
Thinking like engineers...
- Fractals appear in nature and society, but can we start with fractal-like things to do something useful? Sure can! Here are two examples:
/Data compression (image compression being one example, etc.)
- If a fractal is just a simple basic block that gets repeated over and over again, then the only things you need to know about the fractal is what the building block is and how it gets repeated. That's a really efficient way to store a picture of a fractal – much less information than the whole picture. Can we use this idea to store pictures of things which are not fractals? YES! Maybe a picture of something is not a fractal, but something about the picture is fractal-like. If you write a compute program that looks these fractal-like elements and then captures only the essential information – what the building block is and how, it can can be repeated to (approximately) reproduce the original image, then you get a huge savings in how much space you need to store the picture. Inventions like this are part of the reason for why we can have multimedia on the internet today.
/Recursion in computer science.
- Say you're looking through your home for your favorite pencil – what do you do? Open the door, look around, see something of interest: a drawer. Open the drawer, look around, see something of interest: a pencil case. Open the pencil case look around – find your pencil! Close the pencil case, close the drawer, close the room.
- That's an example of a recursive algorithm to solve the search for your pencil. You break up the problem into lots of smaller and smaller pieces, but do the same thing at every level. Lots of problems in computer science can be solved using recursion – it's one of the most useful tools a programmer has. And at its heart is a fractal structure!