Alberto Fernandez-Nieves, Georgia Institute of Technology
As both the colloidal and polymeric properties of the particles are relevant for the behavior of microgel suspensions at high concentrations, their phase behavior is not as well understood as that of hard particles. As a result, it is often the case that theoretical predictions and experimental results largely disagree with each other. For example, theoretical works predict that charged microgels interacting via a Yukawa pair-potential for center-to-center distances above the particle diameter and via an inverse power law potential for distances below the particle diameter should form, at sufficiently high volume fractions, non-cubic crystal structures. However, our previous neutron and X-ray scattering studies on charged microgels as well as experimental results by other groups indicate that the crystal structure is comparable to that of hard spheres even for very dense packing, where the particles have to deform and/or interpenetrate.
Perhaps one of the reasons for this lack of predictability is that even the interaction between microgels and its dependence on particle concentration is still not known in detail: While recent work suggests that microgel particles interact via Hertzian potentials for center-to-center distances slightly below the particle diameter, other works find that this potential does not provide a good model for understanding the phase behavior of this kind of soft particles.
To address the interplay between particle softness and phase behavior, we have performed small-angle X-ray scattering (SAXS) studies of crystal growth in charged microgel suspensions. The microgels are comprised of poly(N-isopropylacrylamide) (pNIPAM), co-polymerized with acrylic acid (AAc). As a result, they are responsive to temperature, by virtue of pNIPAM, and also to pH, as AAc ionizes at pH > 4.3. We study suspensions at a fixed pH little above this value, such that the particles are slightly charged and still show a clear temperature response.
Consistent with what is found in hard sphere suspensions, we find that random hexagonal close packed (rhcp) crystals grow initially. This is a random sequence of close packed hexagonal planes that can be understood as a random mixture of face centered cubic (fcc) and hexagonal close packed (hcp) lattices. The appearance of rhcp crystal shows that the free energy difference between fcc and hcp lattices is small, consistent also with what is expected for hard spheres. Furthermore, the rhcp crystal is found to transform slowly towards the fcc crystal lattice, which appears to be the equilibrium structure, as in hard spheres. However, at intermediate volume fractions, we observed the formation of a body centered cubic (bcc) crystal, which was not stable and, therefore, disappeared as the sample aged.
This crystal structure is expected for hard core Yukawa particles with an intermediate screening length and at volume fractions below those where the fcc structure is the ground state of the system. In contrast, we observe that the bcc transient forms at volume fractions above and below those where we only see formation of rhcp structures that slowly convert to fcc. Interestingly, for core-shell particles consisting of a hard core and a fuzzy corona the bcc structure is expected to form in an intermediate range of volume fractions, between a loosely and a densely packed fcc phase. This behavior results from two competing effects affecting the crystal structure made by this kind of particles. On the one hand, the maximum packing fraction rule, at play in the presence of pure excluded-volume interactions, favors a close packed structure, as the configurational entropy of the system is maximum in this situation. On the other hand, the principle of contact area-minimization, at play since the interaction between the particles scales with the contact area between them, favors formation of more loosely packed structures like the bcc of the A15 lattice, which is a bcc lattice with eight basis atoms in the unit cell. Our results suggest that our observations could be related to the predictions of this model for fuzzy particles.
To ascertain this possibility, we estimated the free energy of fcc and bcc crystals in systems composed of particles with a hard core surrounded by a soft corona. By using representative values of the relevant parameters in the model, we found that the total free energy of the bcc lattice is lower than that of the fcc lattice by ~5%, suggesting that the formation of a transient bcc structure in our microgel suspension could be related to the repulsion between the fuzzy outskirts of the particles and the natural tendency to reduce the associated surface free energy. Despite this fact, we observe that the fcc structure is the one that prevails at all studied volume fractions, suggesting that the entropic contribution to the free energy resulting from the volume excluded to the center of mass of the more massive, central region of the particles dominates the total free energy of the system. Nevertheless, the observation of a transient bcc lattice for volume fractions below and above those where we only see rhcp lattice suggests that the competition between the maximum packing fraction rule and the contact-area minimization principle could be playing a role in our microgel suspensions.