Page 96 - Nanotechnology an introduction

compute the free energy, it should be noted that the relevant temperature is not necessarily what one measures with a thermometer, but one should
only take the (typically small number of) relevant degrees of freedom into account; the temperature is the mean energy per degree of freedom. If
slightly sticky granules are thrown into a rotating drum, or stirred in a mixing bowl, typically they will all eventually clump together to form a “random”
structure (e.g., Figure 8.3(a)), but one which is evidently less random than the initial collection of freely flowing granules, hence the entropy is lower,
but every time this experiment is repeated the result will be different in detail, and one feels that the system is really ergodic if one had enough time
to make all the repetitions, and the ergodicity is only broken because one has insufficient time. To make the example more definite, think of the dry
ingredients of concrete (sand and cement) thrown into a “cement mixer” to which a small amount of water is then added.

Figure 8.3 (a) The result of mixing isotropically sticky cubelets. (b) Putative result of mixing selectively sticky cubelets.

The goal of a self-assembly process suitable for manufacturing devices is for the same structure to be formed each time the constituent particles
are mixed together (Figure 8.3(b))—as was imagined by von Foerster [57]. Of course, energy needs to be put into the system. In the “purest” form
of self-assembly, the energy is thermal (random), but it could also be provided by an external field (e.g., electric or magnetic). If we are satisfied by
the constituent particles being merely joined together in a statistically uniform fashion and, moreover, the process happens spontaneously, then it is
more appropriate to speak of self-joining or self-connecting. The mixing of concrete referred to above is, at least at first sight, an example of such a
self-connecting process; more generally, one can refer to gelation. Approaching the phenomenon from a practical viewpoint, it is clear that gelation
is almost always triggered by a change of external conditions imposed upon the system, such as a change of temperature or dehydration and the
spontaneity implied by the prefix “self-” is absent. The only action we can impose upon the system without violating the meaning of “self-” is that of
bringing the constituent particles together. Note that here we diverge from the everyday meaning of the term “assembly”, which includes the
gathering together of people, either spontaneously as when a group of people wishes to protest against some new measure introduced by an
authoritarian government, or by decree.
If the process in which we are interested is deemed to begin at the instant the constituent particles are brought together, then we can indeed put the
mixing of concrete in the category of self-joining, because we could (although it is not usually done in that way) bring the wetted particles of sand
and cement together, whereupon they would spontaneously join together to form a mass.
The meaning of self-joining (of which self-connecting is a synonym) is then the property possessed by certain particles of spontaneously linking
together with their neighbors when they are brought to within a certain separation. One can also imagine there being kinetic barriers to joining,
which can be overcome given enough time. Note that the particles each need more than one valency (unit of combining capacity), otherwise dimers
would be formed and the process would then stop. A good example is steam condensed to form water. We can suppose that the steam is first
supercooled, which brings the constituent particles (H O molecules) together; the transition to liquid water is actually a first order phase transition
2
that requires an initial nucleus of water to be formed spontaneously. “Gelation” (cf. Section 3.7) then occurs by the formation of weak hydrogen
bonds (a maximum of four per molecule) throughout the system.

Strictly speaking, it is not necessary for all the constituent particles to be brought together instantaneously, as implied in the above. Once the
particles are primed to be able to connect themselves to their neighbors, they can be brought together one by one. This is the model of diffusion-
limited aggregation (DLA). In nature, this is how biopolymers are formed: monomers (e.g., nucleic acids or amino acids) are joined sequentially by
strong covalent bonds to form a gradually elongating linear chain. The actual self-assembly into a compact three-dimensional structure involves
additional weak hydrogen bonds between neighbors that may be distant according to their positions along the linear chain (see Section 8.2.11);
some of the weak bonds formed early are broken before the final structure is reached.
In the chemical literature, self-assembly is often used as a synonym of self-organization. A recapitulation of the examples we have already
discussed shows, however, that the two terms cannot really be considered to be synonymous. The diffusion-limited aggregate is undoubtedly
assembled, but can scarcely be considered to be organized, not least because every repetition of the experiment will lead to a result that is
different in detail, and only the same when considered statistically. “Organized” is an antonym of “random”; therefore, the entropy of a random
arrangement is high; the entropy of an unorganized arrangement is low. It follows that inverse entropy may be taken as a measure of the degree of
organization; this notion will be further refined in the next section. The diffusion-limited aggregate differs from the heap of sand only insofar as the
constituent particles are connected to each other. An example of organization is shown in Figure 8.3(b). The impossibility of self-organization has
been proved by Ashby, as will be described in Section 8.2.3.
Before discussing self-organization, we must first discuss organization, of which self-organization is a part. If the elements in a collection (here we
shall not say “system”, because that already implies a degree of organization) are organized to some degree, that implies that they are in some
way connected to each other, which can be considered as a kind of communication, and are hence subject to certain constraints. In other words, if
the state of element B is dependent on the state of A to some extent, then we can say that B's state is conditional on that of A. Likewise, the relation
between A and B may be conditional on the state of C. Whenever there is conditionality, there is constraint: B is not as free to adopt states as it
would be in a totally unorganized system [10].