Figure 5.10 The distinction between concentration difference and concentration gradient. a: The standard model of diffusion involves a flux, J, of a substance across a barrier of thickness xand cross-sectional area A between two compartments, I and II. b: Concentration gradient, (C; - Cjj)/x, can be altered independently of the concentration difference simply by changing the boundary's thickness.

Obviously, the flux, J, varies directly with the concentration difference. Note that J can be either positive or negative: the sign of the number simply indicates the direction of the flux, whether it is moving from compartment I to II (if Q is greater than Cn) or vice versa (if CI is less than CII). Note as well that there will be no flux if the concentration difference is nil (which happens to be the state of maximum disorder).

The relationship between concentration difference and flux driven by diffusion can be expressed in one of nature's fundamental laws, Fick's law of diffusion. For the case of the box separated into compartments, Fick's law is stated as follows:

where A is the surface area of the permeable barrier separating the compartments, x is the thickness of the barrier, and D is a diffusion coefficient that depends upon the size of the molecule and how it interacts with the barrier.3

Note that I have set off two of the terms in Fick's law, (C: — Cn) and x, with brackets. I have done so because their quotient is a very important quantity known as the concentration gradient (Fig. 5.10). This is not the same as the concentration difference used in the proportionality equation 5.3: the gradient expresses the steepness of the difference. The distinction is easily illustrated by an analogy: the height of a loading platform is one thing, the steepness of a ramp leading up to it is another. Here is the crucial part, so read carefully: Fick's law states that the flux rate is directly proportional to the steepness of the concentration gradient, and not necessarily to the magnitude of the concentration difference. Indeed, one can alter the flux independently of the concentration difference simply by altering the thickness of the barrier separating the two compartments. Make the barrier thin (steepen the concentration gradient) and flux increases, even if the concentration difference is unchanged.

The distinction between a gradient and a difference offers us a clue about how the shape of an accreting surface can change as it grows. At the heart of this

3. Fick's law is more generally (and properly) expressed as a differential equation: dJ = -DdC/dx. This be solved for various geometric configurations—for example, across a flat barrier with thickness x and surface area A. The two-compartment model outlined in the text is the simplest, where flux is uniform and in one direction. Diffusion in different configurations, such as diffusion through tubes or in open space, can be solved by integrating the general form of Fick's law.

mechanism is a simple principle: in a system where diffusion limits accretive growth, local rates of growth will be proportional to the local concentration gradient driving flux of the diffusing material to the surface. This principle can be developed into a general model for accretive growth, known as diffusion-limited accretion (DLA), which is particularly suited to modeling the growth of modular organisms. Let us now flesh out this model with an example.

Assume that a surface, say a layer of coral polyps, grows by accretion of some substance, say calcite, and that the delivery of new material is slower than the accretion rate (the "settling-in time"). In other words, diffusion rate limits the growth by accretion. To understand growth by diffusion-limited accretion, we need to understand in some detail the movements of calcium through the liquid in contact with the accreting surface. Imagine a thin layer of solution above the surface (Fig. 5.11). As calcite accretes to the surface, calcium moves from solution to the surface at a rate Jso^surf. As calcium leaves the thin layer of solution, its concentration there declines. This sets up a concentration difference between the thin layer of solution adjacent to the surface and the layer above it, and this difference drives a flux, Jdiff, downward into the layer of liquid adjacent to the surface. At equilibrium, the concentration of calcium in the surface layer will settle down to some low value, forming an unstirred layer of calcium-depleted liquid at the surface. This unstirred layer forms a diffusion barrier for the movement of calcium from solution to the surface.

The picture changes if we now allow water to flow very smoothly over the surface. When water flows over a stationary surface, it slows down and forms a characteristic velocity profile known as a boundary layer. The fluxes of calcium across a boundary layer are substantially different from those across an unstirred layer. When water flows over the surface, it is as if a tile of water were allowed to reside over the surface for a while, and while "in residence" it deposits some quantity of calcium. Once the calcium in this tile is depleted, it is pushed out of the way and replaced by another tile, which brings in a fresh load of calcium.

surface surface solution solution depletion depletion replacement replacement

surface

Figure 5.11 Diffusion-limited fluxes at an accreting surface. a: Material in a thin surface layer (dark shading) is depleted the longer it sits adjacent to the surface (light shading). Convection is analogous to sliding the depleted layer away and replacing it with a fresh layer. b: The concentration of material at an accreting surface is therefore determined by three fluxes: Jsol^surf, the flux of calcium from solution to the surface; Jjiff, the flux driven by difference in concentration in different layers of the liquid; and Jconv, the flux driven by convection.

surface

Figure 5.11 Diffusion-limited fluxes at an accreting surface. a: Material in a thin surface layer (dark shading) is depleted the longer it sits adjacent to the surface (light shading). Convection is analogous to sliding the depleted layer away and replacing it with a fresh layer. b: The concentration of material at an accreting surface is therefore determined by three fluxes: Jsol^surf, the flux of calcium from solution to the surface; Jjiff, the flux driven by difference in concentration in different layers of the liquid; and Jconv, the flux driven by convection.

This process is represented by an additional flux coming in from the side, a convection flux, Jconv (Fig. 5.11). Concentrations of calcium at the surface will, on average, be higher in flowing solution than they would be in an unstirred layer, and the greater concentration will increase the rate of deposition of calcium to the surface. In the end, concentrations of calcium will form a steady-state boundary layer at the surface. Because Jconv is proportional to the flow rate, the concentration boundary layer will parallel the boundary layer's velocity profile (Fig. 5.12). Speeding up the flow, and therefore increasing the velocity boundary layer, will in turn make the concentration gradient in calcium through the boundary layer steeper. In accordance with Fick's law, the higher concentration gradient will sustain a higher accretion flux of calcium to the surface, and the surface will grow more quickly. The opposite sequence of events applies to slowing the velocity.

So far, this model is fairly uninteresting, because the ideal conditions I have posited (perfectly smooth surface, well-behaved flows and concentration differences) will promote only uniform growth at the surface. Nothing has yet been presented that would promote a change of shape of the surface, which is what needs explaining. Things become interesting, however, when we look at what must happen at a surface imperfection that elevates some parts of the surface above their surroundings.

Describing what happens is easiest if flows over the surface are depicted with streamlines. A streamline simply graphs the trajectory followed by a small parcel of water as it flows. Streamlines themselves do not indicate the parcel's velocity. For example, streamlines above a perfectly smooth surface will form an array of parallel, evenly spaced lines (Fig. 5.13), even though concentration of X velocity of flow 0

concentration of X

concentration of X velocity of flow 0

concentration of X

height above substratum

Figure 5.12 Velocity and concentration boundary layers at an accreting surface. With slow flow (left), the velocity and concentration profiles are similar. If flow increases (right), the concentration profile becomes steeper as well.

height above substratum

Figure 5.12 Velocity and concentration boundary layers at an accreting surface. With slow flow (left), the velocity and concentration profiles are similar. If flow increases (right), the concentration profile becomes steeper as well.

the velocity in the boundary layer changes with height (Fig 5.12). Streamlines sometimes indicate changes of velocity, however. If we introduce an imperfection into the surface, let us say a hemispheric bump, the streamlines will be crowded together above it (Fig 5.13). This crowding together indicates that the velocity of water above the bump has increased, in just the same way velocity of water in a hose is increased when it passes a partial blockage, like a sprinkler nozzle.

Streamlines packed together in this way indicates that the velocity boundary layer above the bump has steepened, and if accretion is happening, the concentration boundary layer will have steepened as well. You can now see the really interesting part: accretion growth will be fastest at the bump, because the concentration gradient driving accretion flux will be steepest there. This has the interesting consequence of magnifying small imperfections in an accreting surface (Fig. 5.13). In short, diffusion-limited accretion changes the shape of an accreting surface by amplifying existing imperfections of shape.

Diffusion-Limited Accretion in Sponges and Corals DLA growth models have been applied to all sorts of fractal growth systems, ranging from the growth of crystals to development of lightning bolts. They have also been used to explain the patterns of growth and form among the sponges and corals, but with an interesting twist. Obviously, accretive growth in, say, a coral is not simply a matter of crystalline enlargement: as we saw in Chapter 2, the deposition of calcite in a corallite requires metabolic energy in the form of ATP. Consequently, the growth of a corallite will depend not only on the rate at which calcium ions and carbonate ions can be delivered to the calciloblast, but also on the rate at which metabolic energy can be delivered there. This rate, in turn, is determined by the rate at which a coral can capture food from its environment. The situation is similar in a growing sponge, because the addition of new spicules or spongin fibers and the cells to fill the spaces between them also requires energy. Although Fick's law of diffusion does a. concentration of X

Figure 5.13 Magnification of surface imperfections at an accreting surface. a: In a flow field with uniformly spaced streamlines, the concentration boundary layer is uniform over the entire surface. b: A surface imperfection, like a small hump, compacts the streamlines over the surface, and the concentration gradients (represented by the differential expression dCx/dh, where h = height) at the hump are steepened locally. c:The locally-steep boundary layers accelerate accretion at the imperfection.

Figure 5.13 Magnification of surface imperfections at an accreting surface. a: In a flow field with uniformly spaced streamlines, the concentration boundary layer is uniform over the entire surface. b: A surface imperfection, like a small hump, compacts the streamlines over the surface, and the concentration gradients (represented by the differential expression dCx/dh, where h = height) at the hump are steepened locally. c:The locally-steep boundary layers accelerate accretion at the imperfection.

not strictly apply, because capture of food particles by a filter feeder like a coral polyp is not really diffusion, the two processes are similar enough that a DLA model should still work. Remember the justification for the electrical analogy in Chapter 3—most rates of energy transfer operate through fundamentally similar processes. Fick's law is, at root, an equation that describes how potential energy does work.

Let us see how one of the common morphological features of sponges and corals, a branch, can be explained by a DLA growth model. Suppose that on a surface comprising several modules there is a slight imperfection. That is, the surface will consist of areas (bumps) that are elevated above other areas (valleys). Because of the steeper "concentration gradients" in nutrients that develop above the bumps, the modules on the bumps will capture more energy and power more growth than will the modules located lower down, in the valleys. The bumps will grow, magnifying the difference between their position and that of the valleys. If a growing bump gets large enough, imperfections in it may become large enough to initiate growth of new branches. The repetition of this process results in the development of the arborescent form common among sponges and corals.

The variation of form seen between sponges and corals inhabiting still water and those inhabiting flowing water also can be explained by a DLA growth model. Cylindrical bodies grow in two dimensions. Longitudinal growth at the ends elongates the cylinder. Radial growth at the outer surfaces makes the cylinder wider. What form the cylinder takes as it grows, either long and thin or short and squat, depends upon the relative rates of growth of these two dimensions. Obviously, if the rate of longitudinal growth is greater than the radial growth rate, a cylinder will grow to be long and thin.

A cylinder standing upright in a boundary layer will distort the streamlines of flow in two dimensions, vertical and horizontal. Distortion in the vertical dimension, as has already been illustrated, will promote elongation of the cylinder. Distortion of flow in the horizontal dimension will promote radial growth of the cylinder. In an environment with no flow, or slow flow, the greatest disturbance of the streamlines will occur at the highest points on the sponge, and elongation growth will be favored. The result will be a long and spindly sponge. At higher flow velocities, the boundary layer is thinner and the concentration gradients are steeper. Nutritive water will be brought to points lower on the cylinder, promoting their growth as well as growth at the tip. The result will be a shorter, squatter morphology.

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