Condensed-matter physics

Melted by mistakes

Two-dimensional polymers are potentially useful structures — if we could only understand their properties. Observations of one polymer's intricate, two-stage, melting transition may help us do just that.

Can the polymers from which moulded shoe-soles are made assemble themselves into two-dimensional repeating structures accurately enough to create materials for certain magnetic and semiconductor applications? Such an advance could replace the expensive electron and X-ray-lithography techniques needed for these uses. Writing in Physical Review Letters, Angelescu et al.1 contribute to this vision with experiments that further our understanding of the ordering processes of certain polymers. They also test a theory that the melting of two-dimensional crystals results from heat-induced defects in their structure.

Generally, thin films of block copolymers — polymers comprising blocks of at least two different molecular chains joined covalently — are cast and heated to form a disordered melt monolayer. This monolayer can be cooled to create ordered patterns, on scales of 20–50 nanometres, that can be replicated in under-lying layers of inorganic material2. In films of the ‘A–B diblock’ copolymers investigated by Angelescu et al.1, spherical domains of polymer B are surrounded by continuous domains of polymer A (Fig. 1a). In thick sections, this structure forms a regular array below an order–disorder transition temperature of 121 °C; above this temperature, the packing is liquid-like. When applied as a 30-nanometre-thick film to a silicon oxide substrate, the polymer diblocks assemble into a hexagonal array of spheres of B above a brush-like layer of A and B.

Figure 1: The A–B diblock copolymer used by Angelescu and colleagues1.
figure1

a, Each molecule consists of a long block of a polymer A (blue) joined covalently to a short block of polymer B (red). In thick sections, the copolymer self-assembles into a ‘body-centred-cubic’ structure of spherical domains of B surrounded by a continuous domain of A. In the thin (30 nm) film studied, the B spheres organize themselves into a single layer, hexagonally packed with an average spacing of 25 nm on a ‘brush’ of A–B chains. b, In a perfect two-dimensional hexagonal array, each sphere has six nearest neighbours. A ‘dislocation’ consists of two extra close-packed rows of spheres (green lines). These terminate at a sphere with only five neighbours, adjacent to a sphere with seven neighbours — a bound pair of ‘disclinations’. A sphere with only five nearest neighbours is missing a 60° wedge of spheres (a−60° disclination); a sphere with seven neighbours has an extra 60° wedge (a+60° disclination). In the hexatic phase, dislocations are generated thermally, but there are no free disclinations. When the hexatic phase melts, the disclinations separate (‘unbind’) to form the disordered liquid.

The melting of analogous two-dimensional arrays consisting of atoms (xenon on graphite, for example), liquid crystalline molecules or colloids has been investigated over the past two decades, driven by the development of the KTHNY theory3,4,5. This theory — named after the initials of its originators — predicts that the transition from crystal to liquid can be split into two transitions. These transitions, which occur at different temperatures, are associated with the formation of isolated defects known respectively as dislocations and disclinations (Fig. 1b). The intermediate, ‘hexatic’ state has previously been identified by direct imaging in two-dimensional arrays of block copolymers6, magnetic bubbles7 and most8,9,10, but not all11, two-dimensional colloids. In this state, the correlation between the orientations of the ‘bonds’ of one domain and those of other domains decays only slowly, as a power law, with the distance between the domains. (In the crystal, this orientation correlation does not decay, whereas in the liquid it decays over a few particle diameters.)

In the KTHNY theory, order in the two-dimensional array decreases continuously through both melting transitions — in contrast to the discontinuous transition observed in three-dimensional crystals. But some results from experiments and simulations challenge this prediction. In two-dimensional colloid arrays in which a particle-density gradient has been set up to probe the melting process, both continuous8 and discontinuous9 transitions have been observed, apparently depending on whether the interaction potential between the colloidal particles is long- or short-range.

So what of block copolymer spheres? Angelescu et al.1 placed their films in a temperature gradient and used scanning force microscopy to image the degree of order in the array directly, over a range of temperatures that encompassed the hexatic-to-liquid transition. They monitored the changes in the orientation correlation length, ξ — a measure of the distance in the liquid over which the bond orientation around a given sphere is correlated with that of the spheres surrounding it. They found that ξ increases slowly as the temperature, T, decreases towards Ti, the temperature of the hexatic–liquid transition (Fig. 2). As T decreases further, it seems to jump rather abruptly; at the same temperature, the density of disclinations drops to zero.

Figure 2: Good fit?
figure2

Graph of correlation length ξ against temperature T from data obtained by Angelescu et al.1 (open circles). The constant values at lower T represent lower limits imposed by the finite scan size. The solid line is a fit to the smooth, exponential transition predicted by KTHNY theory: ξ exp(B/(T−Ti)1/2) with the constant B=4.28 K1/2 and the transition temperature Ti=396.5 K.

Such abrupt jumps are not obviously compatible with the smooth increase in order predicted by theory. But KTHNY predicts4 that the correlation length increases exponentially with (TTi)−1/2 as the film is cooled to Ti. This functional form, though continuous, is very different from the power-law divergence that is usually found for continuous phase transitions (for example, the demixing of a two-component fluid). The prediction also fits Angel- escu and colleagues' data1 near the transition to within the experimental scatter (Fig. 2). So it seems that this aspect of the KTHNY theory escapes its nemesis for now. Given the ‘hockey-stick’ form of the increase in KTHNY correlation length near the transition, it will be difficult to devise an experiment that would conclusively support either the abrupt-jump or exponential interpretation.

This experimental ambiguity should not obscure the fact that two-dimensional block copolymer layers show clear evidence of a hexatic phase between the solid and liquid states — an essential prediction of the KTHNY theory. Crucially, the existence of the hexatic phase is likely to lead to new methods for preparing block copolymer monolayers with better order: the straight edges of channels confining a monolayer induce an isotropic liquid to freeze to the hexatic state with a well-defined orientation near the edge12.

Other types of block copolymer monolayers also melt differently from three-dimensional crystals. Cylinders lying parallel to the substrate, for example, are predicted13 to experience the thermal generation of dislocations, whose disclinations also separate from each other on melting, a prediction recently verified experimentally14. No experimental evidence is yet available, however, on whether the melting transition to the isotropic liquid is continuous, as predicted by theory.

The mechanisms of defect removal as the copolymers cool may dictate whether two-dimensional block copolymer films can be made with sufficient control over their patterning to be useful for lithography. The results of Angelescu et al.1 highlight the importance of eliminating thermally generated defects during the cooling process. Their experiments15, as well as simulations16, have already provided valuable insights into the mechanisms controlling this kinetics. But even when expensive lithography is required, two-dimensional block copolymers could play a critical role in smoothing the roughness of pattern edges17.

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Kramer, E. Melted by mistakes. Nature 437, 824–825 (2005). https://doi.org/10.1038/437824b

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