Dispersive and non-dispersive growth in ice crystals
I have wanted to do a post on the growth of crystals for a while. Although on a totally different size scale (microns) and time scale (hours) than that of oil reservoir growth (million barrels and eons/ages), the essential behavioral notions behind the two cases of growth remain much the same.
I suggest that in the most disordered environments, the role of entropy overrides other factors enough so that some simple dispersion arguments can explain the size distribution completely.
Take as an example the formation of ice crystals in a cirrus cloud. Depending on the surrounding temperature a crystal nucleates on some foreign particle and then starts growing. The atmospheric conditions have enough variety that the growth rate will disperse to the maximum entropy amount given a mean rate value. The end state for volumetric growth will also show the same amount of variation. Put these two factors together and we end up with the same size distribution as we have for oil reservoir sizes.
p(x) = S/(S + x)2where x is the size variate and S is the mean size. This becomes the entroplet form of a probability density function. Plotting this on a logarithmic size scale, the envelope looks the following. I roughly plotted how the dispersed velocity components play into the aggregation of the full profile (note that this serves as a mirror image of the other way of thinking about growth, that of varying end-states).
In some sense, the entroplet aggregates the non-dispersed rate functions which show individually much steeper exponential declines. Graphically, these individual curves do not stand-out on their own since the entropic disorder smooths and disperses the curves efficiently. Keep this in mind for a moment.
The following particle size distribution (PSD) graph shows measurements taken from high altitude cloud experiments. The size gets measured along a single length dimension and the density of the particles takes the place of a probability. I assume that they have plotted the density as a mass function along the x-axis. I convert this to a volumetric size growth problem, integrate the growth for a time duration corresponding to the cloud formation period, and plot the entropic dispersion model result below. The value for S is 2 and the integration scale is 16 (see Note ) .
Particle Size DistributionsThe data fits the entropic dispersion model nicely (green line), but notice at low density that an extra mode shows up as the blue line. This clearly has a sharp exponential drop so likely has a non-dispersive origin. In terms of the higher density entropic model, this stands out as an ordered nucleation regime in the midst of a sea of disordered ice crystal growth modes. Compare to the first figure again and one can see how a distinct peak could occur.
Cloud ice particle number density n(D) vs. the long dimension of particles as observed at the temperature range of -25° C and -30° C [from Platt, 1997]. It shows a bimodal structure in the ice crystal distribution with the second peak at ~500 mm (edit: microns).
Why or how physically can this bump occur? One can understand this in the context of a completely unrelated analysis, that of marathon finishing times. In a race composed of ordinary citizens, mixed in with some elite runners (bribed by prize money), the elites will form a low density bump in a histogram of finishing times.
The incentives and training and good genetics of the elites separate them from the recreational athlete enough so that they generate a statistically measurable deviation from the trend. Anyone who follows sports understands how this can happen.
I assert that an ice crystal growth model could show this same behavior. Some unknown nucleation process has provided an optimal growth environment for these crystals to deviate from the entropic distribution. Hypothesizing, this could take the form of a catalyst or an accommodating growth substrate. With a power tail of -3/2 this might well have a strong diffusive growth component. However, the nuclei occur rarely enough so they do not drown out the much more common random or spontaneously occurring growth centers. It thus shows up as a clear non-dispersive growth mode in a sea of non-uniformity.
On a micro-level, we do have a variety of reproducible structured shapes to bind against. I would predict that nothing like this would ever happen on the scale of oil reservoirs, black swans notwithstanding. On the scale of oil reservoirs, no two substrates will ever have a common origin, so that the entropic trends will dominate.
BTW, I hope this original analysis may prove of some help to those looking at cloud-based climate change forcing functions, or particle size distributions of volcanic ash (see right). It opens up some possibilities to thinking in a different way. From the plot on the Wikipedia page, they use a log-normal fit to the data yet one that uses an entropic dispersion formulation with the appropriate volume/diameter exponent often can work just as well. Below, I use a root 1/2 on volume on dispersive growth which may indicate a diffusion-controlled rate.
 The approach described here, which essentially broadens the peak but retains the power law tail. Convert the linear growth to a time varying parameter (x=kt) and then treat the growth as an evolving pattern which starts from multiple points in time as the cloud develops.
P(x) = Average 1/(1+S/(k*t)) from t=T to t=T+x/kThis is a cumulative so take the derivative with respect to x to get the PDF.
p(x) ~ 1/(S+x) - 1/(S+kT+x)As described in the reference, this broadening also applies to species diversification and for oil field growth. The only complicating factor in this analysis is that oil reservoir is a volume, yet crystal sizes get reported as a length and we have to convert that to a volume. This means the derivative has to include a chain rule to convert the volume x to a length parameter L, x ~ L3 generates dx/dL ~ L2.