Page 35 - Nanotechnology an introduction

these earlier bonds must be broken to allow connexions between points more distant along the chain to be made. Since, as hydrogen bonds, they
have only about one tenth of the strength of ordinary covalent bonds, they have an appreciable probability of being melted (broken) even at room
temperature. Furthermore, the polymer is surrounded by water, each molecule of which is potentially able to participate in four hydrogen bonds
(although at room temperature only about 10% of the maximum possible number of hydrogen bonds in water are broken—see also Section 3.8).
Hence there is ceaseless competition between the intramolecular and intermolecular hydrogen bonds.
3.6. Cooperativity
When considering the interactions (e.g., between precursors of a self-assembly process), it is typically tacitly assumed that every binding event is
independent. Similarly, when considering conformational switches, it has been assumed that each molecule switches independently. This
assumption is, however, often not justified: switching or binding of one facilitates the switching or binding of neighbors, whereupon we have
cooperativity (if it hinders rather than facilitates, then it is called anticooperativity). A cooperative process can be conceptualized as two
subprocesses: nucleation and growth. Let our system exist in one of two states (e.g., bound or unbound, conformation A or B), which we shall label
0 and 1. We have [149]

(3.28)
and

(3.29)
−1
Let {1} = θ denote the probability of finding a “1”; we have {0} = 1 − θ. The parameter λ is defined as the conditional probability of “00” given that
we have a “0”, written as (00) and equal to {00}/{0}. It follows that (01) = 1−(00) = λ−1)/λ. According to the mass action law (MAL), for nucleation we
have

(3.30)
from which we derive (11) = S/λ, and hence (10) = 1 − (11) = (λ − S)/λ. Similarly for growth

(3.31)

Solving for λ gives

(3.32)

To obtain the sought-for relation between θ and S, we note that θ = {01} + {11} = {0}(01) + {1}(11), which can be solved to yield

(3.33)

Cooperativity provides the basis for programmable self-assembly (Section 8.2.8), and for the widespread biological phenomenon of “induced fit”,
which occurs when two molecules meet and “recognize” each other, whereupon their affinity is increased.
3.7. Percolation
Percolation can be considered to be a formalization of gelation. Let us consider the following. Initially in a flask we have isolated sol particles, which
are gradually connected to each other in nearest-neighbor fashion until the gel is formed. Although the particles can be placed anywhere in the
medium subject only to the constraint of hard-body exclusion, it is convenient to consider them placed on a two-dimensional square lattice (imagine
a Gō board). Two particles are considered to be connected if they share a common side (this is called site percolation). Alternatively, neighboring
lattice points are connected if they are bonded together (bond percolation). In principle the lattice is infinite but in reality it may merely be very large.
Percolation occurs if one can trace the continuous path of connexions from one side to the other. Initially all the particles are unconnected. In site
percolation, the lattice is initially considered to be empty, and particles are added. In bond percolation, initially all particles are unconnected and
bonds are added. The problem is to determine what fraction of sites must be occupied, or how many bonds must be added, in order for percolation
to occur. In the remainder of this section we shall consider site percolation. Let the probability of a site being occupied be p (and of being empty, q
4
= 1 − p). The average number of singlets per site is n (p) = pq for the square lattice, since each site is surrounded by four shared sides. The
1
2 6
average number of doublets per site is n (p) = 2p q , since there are two possible orientations. A triplet can occur in two shapes, straight or bent,
2
and so on. Generalizing,
(3.34)
where g(s,t) is the number of independent ways that an s-tuplet can be put on the lattice, and t counts the different shapes. If there is no “infinite”
cluster (i.e., one spanning the lattice from side to side) then

(3.35)
The first moment of this distribution gives the mean cluster size

(3.36)