Birthday Borax and An Explanation of Crystal Nucleation

For my birthday a while back, a friend of mine gave me food coloring, a box of borax and spools of thread. I asked what they were for, and she said crystals.

Fig. 1: Borax crystal growing birthday gift (orig.)

Of course! I hadn’t ever seen people grow borax crystals before, only sugar or salt, so I looked up a tutorial on YouTube. The procedure was pretty standard: heat water up to near boiling, dissolve the borax, insert a pipe cleaner or thread and wait overnight for the solution to cool down and precipitate out crystals. But while the procedure is simple, the science behind nucleation is more complicated and pretty interesting.

In order for a crystal to form in solution, molecules of a substance must conglomerate to an adequate size to encourage the spontaneous coordination of more molecules. A cluster of this adequate size is called a "nucleus," and the process of its formation is called "nucleation." Clusters that are not large enough to be nuclei are called "embryos." The nucleation process can be described in terms of Gibbs free energy, which accounts for both enthalpy, related to the nucleus’ internal energy, and entropy. Gibbs free energy is given by the equation

                      1.       ΔG=ΔH-TΔS (H is enthalpy, T is temperature in Kelvins and S is entropy)

For a newborn crystal born homogenously (meaning suspended in a medium without contact with other surfaces), there are two main energy changes that are occurring. The first is a lowering of molecular energy due to the formation of attractions between coordinating substance molecules for reasons described in Why Cold Drinks "Sweat". This change in energy can be describe by the equation

                  2.       ΔG=VΔG_v (V is volume, ΔG_v is the change in internal energy per unit volume)

To find ΔG_v, we will assume that the crystal-solution system is cooled to slightly below the substance's melting point, T_m. At this small undercooling, ΔH and ΔS can be approximated as temperature independent [1]. But first, at T_m the difference in free energy ΔG between a substance’s solid and liquid forms is 0. Therefore,

                                                                      3.       ΔG_m=ΔH-T_mΔS=0

                                                                      4.       ΔS_m= ΔH/T_m

Now, assuming that ΔH and ΔS are temperature independent, we can use the same expression for entropy as used at T_m to find the value of ΔG_v. While we’re at it, let’s assume the crystal nucleus is spherical for simplicity’s sake. The following are therefore true:

          5.       ΔG_v=ΔH_fv-TΔS= ΔH_fv-T(ΔH_fv/T_m)= ΔH_fv(T_m-T)/ T_m= ΔH_fvΔT/T_m (from equations 1 and 4)

          6.       ΔG=VΔG_v=4/3πr³(ΔH_fvΔT/T_m) (from equations 2 and 5, V_sphere=4/3πr³)

The second energy change occurring is an increase in a newborn crystal’s energy because the surface of the crystal is disrupting bonding in the liquid medium around it. This change can be described as

          7.       ΔG=Aγ_s=4πr²γ_s (A is the nucleus’ surface area, γ_s is the surface free energy value
           characteristic of the medium-solid interface)

This expression is just the disruption free energy per unit area multiplied by the surface area of a sphere. Together, these two energy changes dictate nucleation. Putting the two expressions together, we get

          8.       ΔG_hom=VΔG_v+Aγ_s= 4/3πr³ΔH_fvΔT/T_m+4πr²γ_s (ΔG_hom indicates that this equation is for
           homogenous nucleation)

This equation is very useful and can be used to describe nucleation events from borax crystals precipitating to homogenous cloud formation. To tease a little more information out of equation 8, let’s see how ΔG_hom changes in response to a substance clump slowly growing in radius. This rate of change corresponds to the derivative of ΔG_hom with respect to radius r [2];

                                                             9.       d(ΔG_hom)/dr=4πr²ΔG_v+8πrγ_s

The point where ΔG_hom no longer changes with r corresponds to the peak on the following graph of equation 8 and its component parts, equations 2 and 7.

Fig. 2: Free energy of nucleation as radius increases (Materials Science and Engineering, an Introduction)

Equation 2 is shown as a decreasing curve because ΔH_f is negative for crystallization. This makes sense because as a material crystallizes, energy is lost to its surroundings, so ΔH_f must flow out of the system. To find the critical nucleation radius,

                                           10.   d(ΔG_hom)/dr=4πr²ΔG_v+8πrγ_s=0 (from equation 9)

                                           11.   r^*=2γ_s/ΔG_v

Plugging the critical radius into equation 8 and extracting the negative from ΔH_f to avoid confusion, we get

                                  12.   ΔG^*_hom=-4/3π(2γ_s/ΔG_v)³ΔG_v+4π(2γ_s/ΔG_v)²γ_s=16/3πγ_s³/ΔG_v²

This expression describes the magnitude of the energy change needed to get a molecular cluster up to the size of the critical radius r^*, a sort of activation energy [3].

However, homogenous nucleation is relatively difficult compared to heterogenous nucleation, as I’m sure you’ve probably heard. Heterogenous nucleation occurs when a nucleus forms on a surface, such as a dust particle for clouds or thread for nucleating borax crystals. Because of the complexity of finding heterogenous nucleation equations due to complex volume and surface area terms as well as extra surface energy considerations, I will not be posting these calculations. The calculations can be found in reference 3, but the summarized result is that the critical radius in heterogenous nucleation remains the same as with homogenous nucleation while the nucleation free energy lowers and makes the nucleation process more accessible. This is why minimizing dust or rough surfaces is important for growing larger crystals rather than clusters of small ones.

What has been described above is just a small bit of the complexity of crystal growing. Knowledge of nucleation rates and crystal growth is widely used in processes such as tuning metals to have properties fit for specific purposes or growing single-crystal silicon for computer processors, and I’m sure this information will come up again on later posts. But mild difficulty of math aside, it’s cool stuff, huh? As a treat for reading through this post, watch this fun Minute Earth video on the homogenous nucleation of clouds and see if you recognize some of the concepts we discussed.

I’m planning on growing some borax crystals soon, and when I do I’ll likely write an experiment post about it so be sure to come back and check that out. I’ve already had my first few days of classes and so far it seems that I will be able to continue posting once a week, likely on Sunday or Monday. As always, thanks for reading and I’ll be posting new stuff soon!

Why Cold Drinks "Sweat"

With a horrible heat wave hitting the Philadelphia area, it’s good to think cool thoughts. Already feeling the heat last night, I left a coconut water in the freezer with the intent to drink it but forgot and so took it to work this morning frozen solid. I figured since it’s so hot outside and the metal can is a good conductor, it’d probably melt pretty quickly. And while the ice in immediate contact did melt, the inside remained frozen and I had to cut the top open with scissors to eat it. Before I figured this out, the can had already shed a puddle at my desk. Have you ever wondered why it is that cold things "sweat."

Fig. 1: My favorite coconut juice brand, Foco (pinstopin.com)

Most of us are familiar with the concept of condensation, having learned about the water cycle in elementary school. We are commonly taught in elementary that water exists as vapor at hot temperatures, condenses to liquid as the temperature drops and eventually expands (not condenses, as ice has a lower density than water due to hydrogen bonding) into ice as the temperature drops further. In high school, we learn about the ideal gas law and how pressure also affects phase transitions, yielding the phase diagram.

Fig. 2: Phase diagram for water (myhomeimprovement.org)

So from this standpoint, we are all familiar with why cold things "sweat." What else is there to it? While the basic principles stand, there are some other viewpoints from which we can view this phenomenon.

Phase transitions can be viewed as being an equilibrium process, as is demonstrated by the fact that an ice and water mix maintains a 0°C temperature. In such a mix, the ice melting and the water freezing are competing processes that are controlled by environmental factors; if you cool the mix the ice expands, but if heated the ice melts. Additionally, the entire mix must either become ice or water only before the temperature can deviate significantly from the equilibrium temperature of 0°C. What’s cool about this process is that if you track the energy entering the ice and water mix, say a glass of iced coconut water (let's treat this as an ideal glass of pure iced water), we can predict the corresponding phase transitions based on molecular kinetics.

When bonds are formed, whether strong or weak, we know that energy is released as heat. The reverse is true as well, breaking bonds requiring energy. The direction of bond energy transfer can be simplistically remembered taking into account the conservation of energy in a two molecule one-dimensional collision. Say two water molecules are moving towards each other and stick together upon impact. Where did the kinetic energy go? Ignoring molecular vibrations, the energy had to have been released as work, or heat. In order to separate the water molecules, we need to get them to move apart, a.k.a. add work, or heat, to yield kinetic energy. In our glass of iced water, this sort of energy transfer is happening extremely fast and on a large scale, one that can be described by Le Chatelier’s principle since the ice and water form an equilibrium.

Fig. 3: Ice-water equilibrium state (JVC's Science Fun)

Now let’s put the iced water outside on a hot summer Philadelphia day. From experience, we know that the ice will melt and the water will become unappealingly warm. If we track the direction of energy transfer, the higher energy hot air must be donating energy to the lower energy iced water simply because this is the default direction of energy transfer in our universe according to the Second Law of Thermodynamics. The added energy must translate into kinetic energy as temperature is positively correlated to molecular kinetic energy. From our two water molecule system we know that a decrease in water molecule association is predicted, favoring water over ice and vapor over water. This manifests as the ice melting and the water warming and eventually evaporating.  

What has been so far described, however, is only focused on the iced water itself. Let’s change our basis to focus on the hot air along the iced water glass instead (assume the water glass does not hamper kinetic energy exchange between air and iced water). Hot air carries a lot of water since at higher temperatures water enters the vapor phase preferentially according to Le Chatelier’s principle. From the viewpoint of the air, the cold iced water is pulling kinetic energy from it, accordingly cooling the air within a certain range of the glass. Plugging this information back into our two molecule system, the energy must be afforded by reducing the molecular kinetic energy of the air, increasing the probability of water existing in associated groups, i.e. water. And this is why a cold drink sweats in the summer.

Since school is starting up again, I will not be able to post as frequently as I have been during the summer. I will try to post at least once a week, and will probably be doing so during the weekend since this is when time is most available. Please have patience with me on this, and as always thanks for reading!

On The Definition of Sustainable Energy

Let’s finally cover what I meant by truly sustainable energy production.

Listing electrical energy generation methods portrayed as sustainable, many would consider solutions such as solar power, wind turbines, hydroelectric dams, geothermal plants, sun farms, wave generators, nuclear power and nuclear fusion to fit the image of futuristic green power. But while all of the above may be green, not all of them are sustainable. To me at least, it seems that green energy is measured from the present as a bench point: anything that produces less greenhouse emissions or causes less ecological destruction than methods currently employed is considered "green." However, sustainable energy production is a more definite construct that can be described through thermodynamics.

From thermodynamics we know that every work-energy conversion in our universe is irreversible, meaning that energy is lost as heat every time something is done. Taking the planet earth as a giant engine, which is accurate in the sense that most life processes convert available energy into work and expend heat, we can conclude that the earth needs constant energy input in order to continue running. Aside from ambient space radiation produced by other sources, the sun is the only source of energy earth knows consistently from day to day. This is something many of us should experientially understand, contrasting the death of winter when days are short to the blossoming of spring when long sunlight returns.

Fig. 1: Boston sun angle during the seasons, affecting solar energy influx (Science Blogs)

Therefore, in order to be sustainable within the lifetime of our sun, an extremely high life expectancy for planet earth, our energy generation tactics should be geared towards the sun’s life-giving energy. Does this mean that solar is the only way to go? No, absolutely not. The sun’s energy hitting the earth enacts a grand cascade of events that grow wind, help generate tides and keep the water cycle flowing. This extends the definition of sustainable energy generation to include wind power, wave power, and hydroelectric generation.

Another side category of what I would consider sustainable energy sources includes those whose failure would correspond with the end of life on earth. One example is geothermal energy. Sure, eventually the earth’s core may cool down, but there would be bigger problems associated with this scenario than humans running out of electricity, such as the dissipation of the earth’s magnetic field letting a bombardment of solar radiation char the planet. Wind power may also be considered part of this category, as wind is in part generated by the Coriolis Effect that depends on the earth continuing to spin. If the earth stopped spinning, bad things would ensue.

So what is not covered under the definition of sustainable electricity generation that is commonly perceived as such? The top two that really get to me are nuclear energy and fusion energy. The selling point of nuclear energy is that it is clean in terms of greenhouse gas generation. However, clean does not directly translate into sustainable. The input into nuclear reactors is uranium, and lots of it. Uranium is not an uncommon element on earth, but a large 1000MWe nuclear power plant requires about 200 tons of refined uranium to operate for one year [1]. Refining uranium is itself a wasteful process, and mining large amounts of ore has an ecological impact as well. Nuclear fusion is often sold as the sustainable energy solution of the future, but in actuality nuclear fusion requires the same input as the sun does: hydrogen. With so much hydrogen on earth, why would this be a problem? Call me paranoid, but any time someone wants to convert something necessary for life into a luxury item (yes, electricity is a luxury item) and some useless helium, then I begin to worry. And relative to the sun’s supply of hydrogen, earth’s hydrogen bank is chump change. The hydrogen fusion reaction that generates the most energy is that of deuterium (₁H²) with tritium (₁H³) as described by this equation [2]:

                                                        1.  ₁H² + ₁H³ = ₂He⁴ + neutron

If we calculate the mass difference

       2.  m(₂He⁴) + m(neutron) – m(₁H²) – m(₁H³) = 4.002602 + 1.008665 – 2.01412 – 3.016050  =                   5.011267 – 5.030152= -0.018885 amu

then insert this mass difference into Einstein’s mass-energy equation

                  3.  ΔE=Δmc²

                  4.  Δm=0.018885 amu(1.66053892 x 10^-27 kg/amu)

                  5.  E=(3.135928 x 10^-29 kg)(3 x 10⁸ m/s)² = 2.82 x 10^-12 J/He nucleus formed

and convert the result into a meaningful number

       6.  2.82 x 10^-12 J/He(2.778 x 10^-7 kWh/J)(1 He nucleus formed/2H nuclei consumed)=
            3.92 x 10^-19 kWh/H nucleus consumed(6.022 x 10²³ H nuclei/mol H)(1 mol H/((2.014102 +                   3.0160492)/2) g) = 9.39 x 10⁴ kWh/g H

we get that 9.39 x 10⁴ kWh of power can be generated from one gram of hydrogen gas (assumption made that hydrogen composition is half deuterium, half tritium). Considering that the average American consumes 10,908 kWh annually [3] and that the population of America is about 321 million [4], approximately 37,300 kg of hydrogen gas would be consumed annually to support America alone by fusion energy. This is not zero input.

All in all, what I am trying to say is that science has laid out a very specific definition of what sustainable energy truly is, so it is the job of every person to evaluate whether what is being portrayed as sustainable energy generation is truly backed by the facts.

Does your definition of sustainable energy sourcing differ? Let me know your thoughts in the comments below!