Soft Particles Crossing the Glass Transition

John S. Hyatt and Alberto Fernandez-Nieves

I am one of our group's resident light scattering and rheology specialists, and I use both tools to study colloidal microgels both in dilute and densely-packed solutions.

Fig. 1a: 3D cross-correlation dynamic light scattering. Laser light is split into two beams that cross inside the sample and scatter into detectors at the desired measuring angle. We analyze the time-averaged intensity and the cross-correlation function of the time-dependent intensity fluctuations to obtain information about particles' structure and relaxation timescales (the timescale at which the particle changes position), respectively.
Fig. 1b: Rheology. By applying a steady stress (force per area) and measuring the resulting strain (deformation per thickness), we obtain information about the suspension's viscosity and yield stress. By applying an oscillatory stress and measuring the resulting strain, we obtain information about the suspension's shear moduli and relaxation at timescales equal to the inverse of the oscillation frequency.

I work with a system of microgel particles whose structure can be controlled by ambient temperature and pH to achieve volume (and therefore density) changes by a factor of over 100 (see Fig. 1). They are synthesized from N-isopropylacrylamide (which becomes hydrophobic at high temperatures) and acrylic acid (which ionizes at high pH) and crosslinked with poly(ethyleneglycol), a hydrophilic polymer.

Fig. 1a: Radii of the particles as a function of pH for three different temperatures. At pH higher than 6.5, the acrylic acid is fully ionized, and there is no further pH-dependence.
Fig. 1b: Radii of the particles as a function of T for three different pH. At low pH, the particles display the high-temperature deswelling behavior characteristic of NIPAM; at high pH, this behavior is washed out by the ionized acid groups.

These particles are unusual in that they undergo several dramatic changes in their internal structure at different conditions. Fig. 2 shows the form factors (Fourier-transformed radial density functions, obtained via light scattering) of the particles at different conditions. The form factors at low pH are well-described by the polydisperse core-shell model[1] with additional terms accounting for the heterogeneous distribution of polymer inside the particle[2], $$P(q)=\int\textrm{d}R_{core}f\left(R_{core}\right)\left[3\exp\left(-\frac{q\sigma_{surf}}{2}\right)\frac{\sin\left(qR_{core}\right)-qR_{core}\cos\left(qR_{core}\right)}{\left(qR_{core}\right)^3}\right]^2+\frac{I_{1}}{\left[1+\left(q\xi_{het}\right)^2\right]^2}+I_0,\,\,\,\,\,\,\,\,\,\,(\textrm{Eq.}1)$$ and at high pH by the star polymer model[3], $$P(q)=A_1\int\textrm{d}R_{g}f\left(R_{g}\right)\exp\left[-\frac{1}{3}\left(qR_g\right)^2\right]+\frac{A_2}{\mu q\xi_{blob}}\frac{\sin\left[\mu\arctan\left(q\xi_{blob}\right)\right]}{\left[1+\left(q\xi_{blob}\right)\right]^{\mu/2}}.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\textrm{Eq.}2)$$ The integrals include particle polydispersity following a distribution f, in this case a Gaussian. In Eq. 1, the relevant parameters are the radius of the core, the thickness of the surrounding fuzzy shell, the length scale of the heterogeneities, and the intensity amplitudes of the heterogeneous term and constant background. In Eq. 2, the relevant parameters are the radius of gyration, the blob size of the polymer branches inside the particle, μ, a factor related to the internal fractal dimension of the polymer, and the relative amplitudes of the two terms.

Fig. 2a: pH = 3.1.                                                Fig. 2b: pH = 5.                                                     Fig. 2c: pH > 6.5.
Temperature increases from black points (14 ° C) to pink points (60 ° C). Those two temperatures are broken into two ranges: at lower q are the results from light scattering; at higher q are the results from neutron scattering. (The gap between them is due to the range restrictions on both types of measurement.) Lines are fits to Eqs. 1-3.

We are currently trying to understand the way in which the softness and internal structure of the particles influences their behavior in highly-packed states, where they display a liquid-glass transition at concentrations far above space-filling; in other words suspensions of these particles remain liquid even when they are so tightly-packed that they must compress to fit into the available space. Fig. 3 shows relaxation times vs. ζ, the generalized volume fraction, obtained from a combination of light scattering and rheology measurements.

Fig. 3: The onset of the glass transition for the microgels in the softest, most swollen state, T = 14 ° C and pH > 6.5.

The glass transition is visible as a dramatic increase in the relaxation time. Surprisingly, although the particles are overpacked when the transition occurs, it is quite abrupt, in contrast to earlier work our group has done that found a much more gradual approach to the glass transition for soft particles[4]. Our work with this system is aimed at better understanding the mechanisms link particle softness (a somewhat nebulous term that includes particle interpenetration, compression, and deformation) to the onset of a liquid-solid phase change as the particle concentration is increased. For more information about our lab's work with densely-packed microgel suspensions, visit my colleague Miguel's page here.

References:
[1] M. Stieger, W. Richtering, J.S. Pedersen, and P. Lindner. J. Chem. Phys. 120, 6197 (2004).
[2] J.T. Koberstein, C. Picot, and H. Benoit. Polymer 26, 673 (1985).
[3] L. Willner, O. Jucknischke, D. Richter, J. Roovers, L.-L. Zhou, P.M. Toporowski, L.J. Fetters, J.S. Huang, M.Y. Lin, and N. Hadjichristidis. Macromolecules 27, 3821 (1994).
[4] J. Mattsson, H.M. Wyss, A. Fernandez-Nieves, K. Miyazaki, Z. Hu, D.R. Reichman, D.A. Weitz, Nature 462, 83 (2009).

Soft Condensed Matter Laboratory, School of Physics, Georgia Institute of Technology
770 State Street NW, Atlanta, GA, 30332-0430, USA
Phone: 404-385-3667 Fax: 404-894-9958
alberto.fernandez [at] physics.gatech.edu