Page 99 - Nanotechnology an introduction

8.2.4. Particle Mixtures

Consider the shaking and stirring of a powder—a collection of small (with respect to the characteristic length scale of the final texture) particles of
type A and type B initially randomly mixed up. The interaction energy v is given by the well-known Bragg–Williams expression

(8.7)
where the three terms on the right are the interaction energies (enthalpies) for A with itself, B with itself, and A with B. One would guess that the
mixture is miscible if v < 0, and immiscible otherwise. If the interaction energies are all zero, there will still be an entropic drive towards mixing (the
case of the perfect solution). Edwards and Oakeshott [46] introduced the compactivity X, analogous to the temperature in thermodynamics,
defined as

(8.8)
where V is the volume (playing the role of energy) and S the entropy, defined in analogy to Boltzmann's equation as S = k ln Ω, where Ω is the
B
number of configurations. If the number of particles of type A at a certain point r is (equal to either zero or one), and since necessarily
i
, by introducing the new variable we have m = ±1. Defining , which Bragg and Williams give as
i
[46], three regimes are identified depending on the interaction parameter v/k X[46]:
B
• miscible: v/k X < 1, ϕ = 0;
B
• domains of unequal concentrations: v/k X > 1, ϕ small (X = v/k emerges as a kind of critical point)
B
B
• domains of pure A and pure B: .
8.2.5. Mixed Polymers

The entropy of mixing entities is

(8.9)
where ϕ is the volume fraction of the entities of the ith kind. The per-site (where each site is a monomer unit), free energy of mixing two
i
polymers A and B is

(8.10)
where N and N are the degrees of polymerization, and χ is the Flory–Huggins interaction parameter, given by (cf. equation 8.7)
B
B
(8.11)
where z is defined (for a polymer on a lattice) as the number of lattice directions. The first two terms on the right-hand side of equation (8.10),
corresponding to the entropy of mixing (equation 8.9), are very small due to the large denominators, hence the free energy is dominated by the third
term, giving the interaction energy. If χ > 0, then phase separation is inevitable. For a well-mixed blend, however, the separation may take place
exceedingly slowly on laboratory timescales, and therefore for some purposes nanotexture might be achievable by blending two immiscible
polymers. However, even if such a blend is kinetically stable in the bulk, when prepared as a thin film on the surface, effects such as spinodal
decomposition may be favored due to the symmetry breaking effect of the surface (for example, by attracting either A or B). Separation can be
permanently prevented by linking the two functionalities as in a block copolymer (Section 8.2.6).

8.2.6. Block Copolymers
One of the problems with mixing weakly interacting particles of two or more different varieties is that under nearly all conditions complete
segregation occurs, at least if the system is allowed to reach equilibrium. This segregation is, however, frustrated if the different varieties are
covalently linked together, as in a block copolymer. A rich variety of nanotexture results from this procedure. If A and B, assumed to be immiscible
(χ > 0) are copolymerized to form a molecule of the type AAA⋯AAABBB⋯BBB (a diblock copolymer), then of course the A and B phases cannot
separate in the bulk, hence microseparation results, with the formation of domains with size ℓ of the order of the block sizes (that is, in the
nanoscale), minimizing the interfacial energy between the incompatible A and B regions. One could say that the entropy gain arising from
diminishing the A–B contacts exceeds the entropic penalty of demixing (and stretching the otherwise random coils at the phase boundaries). The
block copolymer can also be thought of as a type of supersphere [136]. If χN is fairly small (less than 10) we are in the weak segregation regime
and the blocks tend to mix; but in the strong segregation regime (χN ≫ 10) the microdomains are almost pure and have narrow interfaces. As the
volume fraction of one of the components (say A) of the commonly encountered coil–coil diblock copolymers increases from zero to one, the bulk
morphologies pass through a well characterized sequence of body-centered cubic spheres of A in B, hexagonally packed cylinders of A in B, a
bicontinuous cubic phase of A in a continuous matrix of B, a lamellar phase, a bicontinuous cubic phase of B in a continuous matrix of A,
hexagonally packed cylinders of B in A, and body-centered cubic spheres of B in A.
When block copolymers are prepared as thin films (thickness d less than 100 nm) on a substratum (e.g., by spin-coating or dip-coating), the
symmetry of the bulk system is broken, especially if one of the blocks of the copolymer is preferentially attracted to or repelled from the surface of
the substratum. If d < ℓ, the surface may be considered to have a strong effect on the structure of the thin film. For example, poly-2-vinylpyridine does
not wet mica, and a polystyrene–polyvinylpyridine block copolymer thin film on mica has a structure different from that of the copolymer in the bulk