## How Nature Uses Rd

Some examples of RD in nature are shown in Figure 1.2. Figure 1.2 Examples of animate (a-d) and inanimate (e-h) reaction-diffusion systems on various length scales. (a) Calcium waves propagating in a retinal cell after mechanical stimulation (scale bar 50 mm). (b) Fluorescently labeled microtubules in a cell confined to a 40 mm triangle on a SAM-patterned surface of gold (staining scheme green microtubules red focal adhesions blue actin filaments scale bar 10 mm). (c) Bacterial colony growth...

## Modeling of 3D RD

The key feature of the etching process is the sharpness of the RD front. To understand its origin, let us first, as usual, write the equations governing the RD process. Denoting the metal by the subscript M, and the etchant by E, we have where CM is the concentration of metal colloids NPs (in terms of atoms) immobilized in the cube, CE is the concentration of etchant (limiting species), D is the diffusion coefficient of the etchant, k is the apparent reaction constant and a and b are the...

## Contents

1 Panta Rei Everything Flows 1 1.1 Historical Perspective 1 1.4 RD in Science and Technology 9 References 12 2 Basic Ingredients Diffusion 17 2.2 Solving Diffusion Equations 20 2.2.1 Separation of Variables 20 2.2.2 Laplace Transforms 26 2.3 The Use of Symmetry and Superposition 31 2.4 Cylindrical and Spherical Coordinates 34 2.5 Advanced Topics 38 References 43 3.1 Reactions and Rates 45 3.2 Chemical Equilibrium 50 3.3 Ionic Reactions and Solubility Products 51 3.4 Autocatalysis, Cooperativity...

## Buckling Into The Third Dimension Periodic Nanowrinkles

We saw in Chapter 6 that the formation of a precipitate inside of a gel matrix can cause gel swelling. This is also the case for PP where the zones of Ag2Cr2O7 precipitate rise above the gel's 'background' level. Interestingly, surface profilo-grams and atomic force microscope images in Figure 7.12 show that the sub-micrometer heights, h, of the buckled bands increase roughly linearly with band position, x. Before we see how this scaling can be used in some nontrivial microfabrication, let us...

## Remote Exchange of the Cores

Once fabricated, the cores can be further modified 'remotely' by galvanic replacement reactions (Figure 10.6). For example, when an agarose cube containing a copper core is immersed in 20 mM aqueous solution of HAuCl43H2O, the following redox reaction takes place 2AuCl47 + 3Cu > 2Au + 3Cu2+ + 8Cl . Since the difference in standard potentials, E0 is positive, this reaction spontaneously transforms the copper colloids in the cores into gold ones. Because the equilibrium constant for the...

## Diffusion In Solids The Kirkendall Effect And Fabrication Of Coreshell Nanoparticles

Although our current repertoire of RD fabrication techniques is quite flexible in terms of the reacting chemicals, it is limited by the required porosity permeability of the supporting medium (a gel, a polymer, or an elastomer). In this section, we will study - and later apply in nanofabrication - phenomena that overcome this limitation with the help of so-called solid interdiffusion. Can atoms of crystalline solids such as metals or semiconductors diffuse into one another Indeed, they can. The...

## Structuring Metal Films

There are many good reasons to microstructure metal films thin metal foils with holes can be used as membranes (Figure 8.9(a)) for separating small particles, disjoint metal Figure 8.9 Etching micropatterns into metals. (a) Array of circular holes etched through a freestanding, 100 mm thick copper foil the holes correspond to disjoint, raised features in the gel stamp. (b) Disjoint metal plates obtained from the same foil the lines along which the foil is 'cut' correspond to the network of...

## Oscillating Reactions

The autocatalytic and or inhibitory reactions are also the key components of nonbiological reaction-diffusion systems producing chemical waves and Turing patterns (cf. Chapter 1). Before we see in Chapter 9 how these fascinating phenomena emerge from the coupling between 'looped' chemical transformations and diffusion, we will first discuss their chemical kinetic component, which gives rise to temporal concentration oscillations in the absence of diffusive flows. A closed chemical system (i.e.,...

## Autocatalysis Cooperativity And Feedback

Not all chemical reactions progress linearly from substrates to products - some can literally loop onto themselves and either speed up or slow down their own progress. In autocatalytic reactions, products accelerate the reaction. As an example, consider a simple reaction A + B 2B, with rate law d A dt k A B .3,4 Noting that A 0 A B B 0 and defining x A , we obtain B A 0 + B 0 A A 0 + B 0 x. After substitution, the rate law becomes dx dt kx( A 0 + B 0 x), which can be integrated to give A 0 + B...

## Governing Equations

Consider an ionic reaction of the form nAm + + mB C. Diffusion of the ions through the gel is described by diffusion coefficients DA and DB, and rapid formation of C occurs when the product of concentrations A B m exceeds the solubility product, Ksp. The created C molecules, however, do not immediately precipitate but are instead free to diffuse (with diffusion coefficient DC) until their local concentration reaches some saturation threshold C *. At that point, nucleation occurs followed by...

## Case 1 tRxn tDiff

When reaction and diffusion take place on similar time scales or when diffusion is faster (i.e., its characteristic time, tDiff, is smaller than that of reaction, tRxn), one may treat the processes simultaneously, using time discretization and integration methods appropriate for diffusion problems. 4.4.1.1 Forward time centered space (FTCS) differencing In the simplest numerical approach, the RD equations may be solved by replacing the time derivative by its forward difference approximation and...

## Solving Diffusion Equations

Mathematically, Equation 2.8 is a second-order partial differential equation PDE , whose general solutions can be found by several methods. As with every PDE, however, the knowledge of a general solution is not automatically equivalent to solving a physical problem of interest. To find the 'particular' solution describing a given system e.g., our test tube , it is necessary to make the general solution congruent with the boundary and or initial conditions. By matching the general solution with...

## Reactions And Rates

Now that the reader is a seasoned expert on diffusion, it is time to explore the world of reactions. As already discussed in Chapter 2, molecules are very dynamic entities, constantly moving and colliding with their neighbors. This 'aggressive' behavior is the basis for chemical reactions, and if the energy supplied by the colliding molecules is enough to break their bonds, they can combine to give new products. Chemical kinetics links these microscopic collisions to the macroscop-ically...

## Self Assembly of Open Lattice Crystals

Leaving other fabrication schemes and applications of individual particles to the creative reader, let us consider the collections of such CSPs. The opportunity here is to combine RD particle fabrication with self-assembly10,11 - that is, the process by which discrete components organize without any human intervention into ordered and or functional suprastructures. The unique feature of CSPs is that their self-assembly leads to structures in which the cores are separated from one another and...

## Galvanic Replacement And Dealloying Reactions At The Nanoscale Synthesis Of Nanocages

As mentioned at the end of the previous section, one of the major uses of hollow metal nanoparticles is in optical detection.21 Due to their small sizes, metal nanoparticles have optical properties very different than those of the corresponding bulk metals - for instance, 5 nm particles of gold are red violet whereas silver particles appear yellow orange. These colors result from the confinement of the electrons within the metal NPs and from collective electron motions - known as surface...

## References

Chu, L.Y., Utada, A.S., Shah, R.K. et al. 2007 Controllable monodisperse multiple emulsions. Angew. Chem. Int. Ed., 46, 8970. 2. Pekarek, K.J., Jacob, J.S. and Mathiowitz, E. 1994 Double-walled polymer microspheres for controlled drug-release. Nature, 367, 258. 3. Kim, S.H., Jeon, S.J. and Yang, S.M. 2008 Optofluidic encapsulation of crystalline colloidal arrays into spherical membrane. J. Am. Chem. Soc., 130, 6040. 4. Nguyen, D., Chambon, P., Rosselgong, J. et al. 2008 Simple route to get...

## Microetching Transparent Conductive Oxides Semiconductors and Crystals19

Etching micropatterns in transparent conducting oxides such as indium-tin oxide ITO or zinc oxide ZnO and in semiconductors e.g., GaAs is of great importance for the fabrication of optoelectronic devices ITO electrodes , sensors and on-chip UV lasers ZnO , as well as integrated circuits, solar cells and optical switches GaAs . Since all of these applications rely on the ability to define pertinent microscopic architectures, a variety of methods have been developed to micropattern these...

## List of Boxed Examples

2.1 Unsteady Diffusion in an Infinite Tube 30 2.2 Unsteady Diffusion in a Finite Tube 31 2.3 Is Diffusion Good for Drug Delivery 37 2.4 Random Walks and Diffusion 42 3.1 More Than Meets the Eye Nonapparent Reaction Orders 46 3.2 Sequential Reactions 49 4.1 How Diffusion Betrayed the Minotaur 68 4.2 The Origins of the Galerkin Finite Element Scheme 74 4.3 How Reaction-Diffusion Gives Each Zebra Different Stripes 89 6.1 A Closer Look at Gel Wetting 106 6.2 Is Reaction-Diffusion Time-Reversible...