# Soft Matter and Clouds

Brandy Pappas - Final Wiki Entry for APPHY 225, Fall 2008

Hath the rain a father? or who hath begotten the drops of dew?

Out of whose womb came the ice? and the hoary frost of heaven, who hath gendered it?

Canst thou lift up thy voice to the clouds, that abundance of waters may cover thee?

Who can number the clouds in wisdom? or who can stay the bottles of heaven.

Canst thou send lightnings, that they may go and say unto thee, Here we are?

Job 38, misc.

From the beginning of the Microphysics of Clouds(1)...

This project began from a tangent in class about cloud electrostatics, in which Dr. Morrison was trying to remember an article that he read on clouds from different areas of the world having varying base level charges. We presented the work that a few of us had fiddled with at the Complex Fluids Workshop held at Harvard this fall, and seemed to get a great reception from it: I sent out quite a few e-mails with the biological ice nucleators paper attached. Dr. Morrison and I talked about doing another, perhaps longer, talk about Clouds as Soft Matter for the March Complex Fluids Workshop, so for my final project I am going to try to cull together what we as a class thought was interesting, add some more fun facts, and try to organize it in a way that is persuasive.

## Introduction: Why could a Cloud be considered Soft Matter?

What is soft matter? Foams, emulsions, gels, all have certain characteristics that cause us to classify them as soft condensed matter. According to RAL Jones(2), in his book "Soft Condensed Matter Physics", there are several points of interest in defining such materials.

Length scale is very important. Typically in soft condensed matter the self-assembled structures are between atomic and macroscopic scales. Many of the behaviors of this type of matter come from the topological implications of the assembled units, and are not necessarily dictated by the chemistry of the units; this is especially true in a heterogeneous substance like clouds. Shown below, we see that the droplet sizes in clouds range from approximately 10um to 3mm(1).

This can be thought consistent with colloidal dispersions, which according to Jones are often but not necessarily submicrometer particles of a solid or liquid dispersed in another material(2). For clouds, there is a wide range of materials present, including ice particles and liquid drops dispersed amongst organic and inorganic debris and, apparently, sometimes even frogs. While frogs may be off of the length scale, overall it can be said that the dispersion of particles is on a similar order to a colloid, and that the final cloud structure is a macroscopic entity of these heterogeneous subunits(3).

Fluctuations in soft matter are also important features, with the typical energies associated with interstructural bonds and with the distortions of those structures comparable in size to thermal energies, kT(2). This is well represented in the graph above from the Microphysics of Clouds and Precipitation referenced below. An interesting example of this property in clouds by one of our classmates shows the water droplets in clouds to be on the order of kT in energy. Additionally, the free energy of coalescence can be shown in be on the order of kT (5), discussed in the microphysics section below.

Soft matter systems are visualized as being in a constant state of motion--the tumbling of cloud over itself, the formation of "cloud highways" and other structures, all indicate that clouds are experiencing some form of long range interactions. Combined, these interactions additionally indicate condensed matter.

Self assembly, too, is a key feature in soft matter, and even more interesting when molecular level assembly (free water droplets coming together around nucleating sites, for instance) leads to supramolecular structure and a higher level of order. Perhaps the reader can imagine a massive cloud as being such a structure; indeed, the field of cloud microphysics is based around this concept. According to Latham(5):

"The microphysics of clouds [...] is concerned with the constituent cloud particles,
their interactions, and their growth to produce rain, snow, hail, and other forms
of precipitation. [T]he overall objective [of cloud microphysics] is defined by
Fletcher (1962 a), as being to explain the behaviour of the cloud as a
macroscopic entity in terms of the microscopic entities that compose it."

The coup de grâce, perhaps, in analyzing clouds as soft condensed matter comes from the discovery that some of the nucleating sites of clouds actually might be much more interesting than originally imagined. In 2008, Brent Christner and his lab at LSU discovered that certain bacterial plant pathogens were ubiquitous ice nucleators, and were in fact necessary and sufficient for the nucleation of ice at warmer temperatures(3)(4). Although a lot of people knew that there were microbes in clouds and throughout the atmosphere, this work, discussed in the Biological Ice Nucleators section below, showed that biological nuclei are the only things that can cause warmer weather clouds to form the droplet sizes they need to begin precipitation.

## Microphysics of Clouds

In keeping with the aspects of clouds formation, this discussion will mainly be regarding phase changes with respect to nucleation sites. According to Latham, three phase changes may be associated with the formation and growth of clouds: the condensation of water vapor to form cloud droplets, the growth of ice from the vapor by means of sublimation, and the freezing of supercooled water droplets(5). All three of these phase changes require nucleation, and "in any phase near the transition point there are microscopic variations in structure, which correspond to the transient appearance of embryos of the neighbouring phase"(5); this is known as heterophase fluctuation. To understand this fluctuation, a study of nucleation processes, the coalescence of each of the phases, is critical.

### Nucleation of Water Vapor

The following is a treatment by Latham on the free energy associated with the homogeneous condensation of water vapor, and further on the heterogeneous aspects associated with condensation on particles:

“In the case of the homogeneous nucleation of water vapor condensation the free-energy change $\Delta G$ associated with the formation of an embryo droplet of radius $r$ in a supersaturated vapor of pressure $p$ at a temperature $T$ is given by

where $n_L$ is the number of molecules per unit volume of liquid, $k$ is Boltzmann's constant, $p_\infty$ is the equilibrium vapor pressure at the temperature $T$ over a plane liquid surface, and $\sigma_{LV}$ is the interfacial free energy per unit area between liquid and vapor.

This free-energy change must be a minimum for a system in equilibrium, and the values of $\Delta G$ and $r$ at this point are determined by setting $\frac{\partial \Delta G}{\partial r} = 0$ in [the following] equation which yields:

for the equilibrium values. [This equation] is the classical formula derived originally by Kelvin (1870).

Droplet embryos with radii $r < r*$ are unstable and tend to disappear under the buffetings of thermal agitation. However, embryos that have exceeded the critical radius $r*$ tend to grow without limit and become macroscopic droplets. It is therefore necessary to evaluate the rate at which embryos of this critical size are generated within the vapor.

A similar procedure can be adopted in calculations of the nucleation rates governing the formation of droplets within atmospheric clouds by means of heterogeneous nucleation on condensation nuclei. In the case of the condensation of water vapor on a plane surface of an insoluble particle, which was first treated in detail by Volmer (1939), the critical supersaturation depends upon the angle of contact phi of the liquid on the substrate, which controls the curvature of the embryo droplet. The critical free energy is found to be

The value of the critical free energy […] is different from that for the homogeneous case […] only by the factor $f(M)$. Since $-1 \leq M \leq 1$ it follows […] that $0 \leq f(M) \leq 1$, so that foreign surfaces have the general property of reducing the free energy necessary to form a critical embryo.”

### Nucleation of Ice

Further treatment of the nucleation of ice by Latham is explored below. This is of particular interest in looking at the temperature associated with the spontaneous nucleation for the bacterial discussion in the next section.

“Because ice is a crystalline structure it is no longer generally sufficient to consider the embryos to be spherical, and a more general polyhedral form must be considered. The exposed faces of the embryo will be those of lowest free energy, and the equilibrium habit can be determined by minimizing the total free energy of an embryo of given size with respect to its habit. Since no definite information is available concerning the free energies of different crystal faces it is necessary to use an average free energy $\sigma_{SL}$ per unit area.

If we assume a particular crystal habit for the embryo then it is convenient to consider an inscribed sphere of radius $r$ so that the volume of the embryo is $\frac {4}{3}\pi r^3 \alpha$ and its surface area $4 \pi r^2 \beta$, where $\alpha$ and $\beta$ are both greater than unity but approach unity as the polyhedron tends to a spherical shape.

Following the procedure adopted [in the previous section] it can be shown that the nucleation rate is given by […]

where $h$ is Planck's constant, $\Delta g$ is the activation energy for self-diffusion across the liquid-solid boundary, and the critical free energy is

where $\epsilon = \frac {\beta^3}{\alpha^2}$, $\Delta T$ is the degree of supercooling, and $\Delta S_{\nu}$ is the entropy of fusion of ice per unit volume. Since the coefficient of $exp(\frac{-\Delta G*}{kT})$ is of order $10^{30} cm^{-3} s^{-l}$ the nucleation rate may be expected to have a very sharp threshold. This prediction is qualitatively consistent with the observation of an extremely well defined freezing threshold close to -40ºC.”

Facilitation of these condensation events, of course, would dramatically increase their probability at high temperatures. This leads into the discussion of bacterial nucleation.

## Biological Ice Nucleators

Biological Ice Nucleators (green)

Ice formation in tropospheric clouds is required for snow and most rainfall. At temperatures greater than –40°C, ice formation is not spontaneous, and diverse substrates can act as catalysts of ice nucleation(3). Biological ice nucleators are the most active ice nucleators in nature, and some bacterial plant pathogens can catalyze ice formation at temperatures near –2°C(4). The activity of most known biological ice nucleators is mediated by proteins or proteinaceous compounds. Results from Brent Christner's lab at LSU indicate that these particles are widely dispersed in the atmosphere, and, if present in clouds, they are likely to have an important role in the initiation of ice formation, especially when minimum cloud temperatures are relatively warm.

Proof of bacterial existence was shown using lysozyme sensitivity (which damages the peptidoglycan cell wall of gram negative bacteria), heat sensitivity, and DNA presence(3). A few microbes were detected in cultured samples, though many were unable to be grown up in the lab (as is sadly the case with some interesting bacterium). The results of the lysozyme and heat sensitivity studies are shown below, with an emphasis on the ubiquity of bacteria in the world-wide samples(3).

Interestingly, the biological aerosols, as Christner calls them, that are most widespread and well studied are those that are associated with plant pathogeneity. These include Pseudomonas syringae, Pseudomonas viridiflava, Pseudomonas fluorescens, Pantoea agglomerans, and Xanthomonas campestris(4). One may wonder whether these bacteria evolved specifically to nucleate clouds, or whether they are merely passengers of ocean aerosolization. Since their hosts are made vulnerable by rain and ice, however, it would appear as though they are indeed opportunistic. This would be quite an evolutionary accomplishment, especially considering the protein technology that the bacteria would have had to create in order to nucleate ice crystals. According to Christner, ice-nucleating strains of P. syringae (which are incredibly pathogenic, attacking plants like beets, wheat, grasses, barley, crabapple trees, peas, etc.) "possess a 120- to 180-kDa ice nucleation active protein in their outer membrane comprised of contiguous repeats of a consensus octapeptide; the protein binds water molecules in an ordered arrangement, providing a nucleating template that enhances ice crystal formation" (4). This a large extracellular protein, especially one that is active (generally using ATP phosphate cleavage as energy), and is likely to have been selected for specifically by the bacteria. In addition to nucleating crystals, this flagellum-like protein is used for motion in liquids (think: water!). A picture is shown below:

Additionally, there are fungi that nucleate ice at low temperatures, as well as some pollens and algae. The fact that these types of organisms and proteins are present in clouds seems like a very dramatic case for the self-assembly and order inherent in soft condensed matter. In light of the size scale of the protein (nanometers in diameter) and the fact that most biological systems require energy on the order of kT in order to unfold membrane bound proteins(2), this argument is made stronger.

## References

1. Microphysics of Clouds and Precipitation by Hans R. Pruppacher and James D. Klett; Springer: New York; 1997
2. Soft condensed matter by R.A.L. Jones; Oxford University Press: New York; 2002
3. Brent C. Christner, Cindy E. Morris, Christine M. Foreman, Rongman Cai, David C. Sands; Science; 29 February 2008; Vol 319: 1214
4. Brent C. Christner, Rongman Cai, Cindy E. Morris, Kevin S. McCarter, Christine M. Foreman, Mark L. Skidmore, Scott N. Montross, and David C. Sands; PNAS; 2 December 2008; Vol 105, 8: 18854
5. J. Latham; Rep. Prog. Phys.; 1969; Vol 32: 69