Absence of crystals in the phase behavior of hollow microgels

Solutions of microgels have been widely used as model systems to gain insight into atomic condensed matter and complex fluids. We explore the thermodynamic phase behavior of hollow microgels, which are distinguished from conventional colloids by a central cavity. Small-angle neutron and x-ray scattering are used to probe hollow microgels in crowded environments. These measurements reveal an interplay among deswelling, interpenetration, and faceting and an unusual absence of crystals. Monte Carlo simulations of model systems confirm that, due to the cavity, solutions of hollow microgels more readily form a supercooled liquid than for microgels with a cross-linked core.

Figure 1

(a) Hydrodynamic radius, $R_h$, vs temperature, $T$ for hydrogenated (MB-HS-105-5-pNIPAM) and deuterated (MB-HS-105-5-D7pNIPAM) 5 mol% cross-linked hollow microgels used in the SAXS and SANS experiments (circles and squares, respectively). (b) Size distribution, $P\left(R\right)$, of the hydrogenated hollow 5 mol% cross-linked microgels (MB-HS-105-5-pNIPAM) at $20{}^{\circ }\mathrm{C}$ and $\theta ={100}^{\circ }$ as obtained from the analysis of the intensity autocorrelation function using a modified Contin algorithm (circles) [26]. The solid curve is a Gaussian fit to the data.

Figure 2

Relative viscosity ${\eta }_r$ vs polymer weight fraction $c$ in solutions of hydrogenated hollow 5 mol% cross-linked microgels (MB-HS-105-5-pNIPAM), circles; deuterated hollow 5 mol% cross-linked microgels (MB-HS-105-5-D7pNIPAM), squares; and regular hydrogenated 5 mol% cross-linked microgels (MB-pNIPAM-5-225), triangles. Solid lines are fits of the data according to Eq. (1).

Figure 3

Photographs of solutions of hollow microgels at different concentrations: (a) MB-HS-105-5-pNIPAM; (b) MB-HS-105-5-D7pNIPAM; (c) MB-HS-60-2.5-pNIPAM. Data relating to the synthesis and characterization of these samples are reported in Table 1 (and Table 3 in Appendix pp1). All samples were stored at $20.0\pm 0.5{}^{\circ }\mathrm{C}$. No crystals are evident up to the highest concentrations. In panels (a) and (b) the samples are contained in vials with dimensions $45\times 14.75$ mm, while in panel (c) the samples are contained in vials with dimensions $32\times 11.6$ mm.

Figure 4

Photographs of solutions of regular hydrogenated 5 mol% cross-linked microgels (MB-pNIPAM-5-225 in Table 1) at different generalized volume fractions $\zeta$. Crystals are visible, from Bragg diffraction, for $0.59\pm 0.01<\zeta <0.70\pm 0.02$. Samples are contained in vials with dimensions $45\times 14.75$ mm and stored at $T=20.0\pm 0.5{}^{\circ }\mathrm{C}$.

Figure 5

SAXS intensities, $I\left(q\right)$, vs scattering vector, $q$, of solutions of hollow 5 mol% cross-linked microgels with outer radius $210\pm 8$ nm and internal cavity radius $91\pm 4$ nm (MB-HS-105-5-pNIPAM in Table 1). Concentrations: $\zeta =0.60\pm 0.01$ (circles), $\zeta =0.75\pm 0.02$ (triangles), $\zeta =0.85\pm 0.02$ (squares), and $\zeta =0.90\pm 0.02$ (diamonds). All measurements were made at $T=20{}^{\circ }\mathrm{C}$.

Figure 6

Structure factors, $S\left(q\right)=I\left(q\right)/I(q,\zeta =0.08)$, vs scattering vector, $q$, of solutions of the hydrogenated hollow 5 mol% cross-linked microgels in Fig. 3 measured with SAXS at different concentrations (blue circles): $\zeta =0.60\pm 0.01$ (a), $0.70\pm 0.01$ (b), $0.75\pm 0.02$ (c), $0.80\pm 0.02$ (d), $0.85\pm 0.02$ (e), and $0.95\pm 0.02$ (f). Light blue solid lines: Gaussian fits of the first peak of the structure factors.

Figure 7

Sketch of a hollow microgel and of the relative polymer volume fraction density profile with the corresponding characteristic length scales.

Figure 8

(a) SANS intensities, $I\left(q\right)$, vs scattering vector, $q$, of hydrogenated hollow 5 mol% cross-linked microgels (MB-HS-105-5-pNIPAM in Table 1). (b) Radial distribution of relative polymer volume fraction obtained by fitting curves in (a) using model of Ref. [23]. Concentrations: $\zeta =0.080\pm 0.003$ (circles), $0.30\pm 0.01$ (left-side triangles), $0.65\pm 0.01$ (squares), $0.87\pm 0.02$ (diamonds), $1.19\pm 0.03$ (upside triangles). All measurements were made at $T=20{}^{\circ }\mathrm{C}$. Colors, symbols, and concentrations in panel (b) correspond to those in panel (a).

Figure 9

(a) SANS intensities, $I\left(q\right)$, vs scattering vector, $q$, of hydrogenated regular 5 mol% cross-linked microgels. (b) Radial distribution of relative polymer volume fraction obtained by fitting curves in panel (a) using model of Ref. [60]. Concentrations: $\zeta =0.080\pm 0.003$ (circles), $0.30\pm 0.01$ (left-side triangles), $0.65\pm 0.02$ (squares), $0.87\pm 0.02$ (diamonds), $1.19\pm 0.03$ (upside triangles). All measurements were made at $T=20{}^{\circ }\mathrm{C}$. Colors, symbols, and concentrations in panel (b) correspond to those in panel (a).

Figure 10

Variation of the radius measured with SANS divided by the radius at $\zeta =0.080\pm 0.003$ and $T=20.0\pm 0.01{}^{\circ }\mathrm{C}$ vs the generalized volume fraction, $\zeta$, for hydrogenated hollow 5 mol% cross-linked microgels (empty circles) and hydrogenated regular 5 mol% cross-linked microgels (solid squares).

Figure 11

Apparent size polydispersity, $p$, vs generalized volume fraction, $\zeta$, for the hydrogenated hollow 5 mol% cross-linked microgels (circles) and hydrogenated regular 5 mol% cross-linked microgels (squares) embedded in the very same matrix of deuterated hollow 5 mol% cross-linked microgels.

Figure 12

SANS intensities, $I\left(q\right)$, vs scattering vector, $q$, of hydrogenated hollow 5 mol% cross-linked microgels with ${\zeta }_H=0.080\pm 0.003$ measured with contrast variation for concentrations, from bottom to top: $\zeta =0.080\pm 0.003$, $0.50\pm 0.01$, $0.65\pm 0.01$, $0.74\pm 0.01$, $0.90\pm 0.02$, $1.00\pm 0.02$, $1.10\pm 0.02$, and $1.19\pm 0.03$. The solid lines represent fits with the hollow fuzzy-sphere model, while the dashed lines represent fits of the data with the ellipsoidal model.

Figure 13

Particle diameter $2R_{\text{SANS}}$ (measured with SANS, empty circles) and nearest neighbor distance $d_{nn}$ (measured with SAXS, solid circles) vs generalized volume fraction $\zeta$ for hollow microgel solutions. Red curve is a fit of the $d_{nn}$ data to the function $c{\zeta }^{-1/3}$ with fit parameter $c$. The vertical dashed lines mark the border of the different compression regimes (i) to (iv).

Figure 14

Snapshot from a Monte Carlo simulation of a solution of soft, compressible, hollow microgels (blue spheres) with concentric cavities (red spheres).

Figure 15

Simulation data for actual volume fraction $\varphi$ vs generalized volume fraction $\zeta$ in solutions of hollow microgels with collapsed outer radius $b_0=63$ nm, collapsed inner radius $a_0=32$ or 50 nm, Flory solvency parameter $\chi =0.2$, and cross-linker fraction $x=0.01$ or 0.001. The straight line, $\varphi =\zeta$, represents incompressible microgels and the horizontal line, $\varphi =0.7405$, represents the close-packed limit for hard spheres. Data are shown also for softer, regular microgels with no cavity ($a_0=0$). Statistical error bars are smaller than symbols.

Figure 16

Simulation data for mean swollen inner radius $a$ and outer radius $b$ (inset) vs generalized volume fraction $\zeta$ for systems of Fig. 15, with collapsed inner radius $a_0=32$ nm (empty circles) or 50 nm (solid circles), and cross-linker fraction $x=0.001$. Statistical error bars are smaller than symbols. The curve is a fit of half the nearest-neighbor distance to the function $d_{nn}/2=c{\zeta }^{-1/3}$ with fit parameter $c$ for $a_0=32$ nm.

Figure 17

Simulation data for bulk thermodynamic properties of hollow microgel solutions: (a) osmotic pressure $P$ and (b) mean pair interaction energy $〈U〉$ vs generalized volume fraction $\zeta$ in solutions with system parameters of Fig. 15, collapsed inner radius $a_0=32$ or 50 nm, and cross-linker fraction $x=0.001$. Empty (solid) symbols represent data from runs initialized in random (fcc crystal) configurations. Statistical error bars are smaller than symbols.

Figure 18

Simulation data for bulk structural properties of hollow microgel solutions: (a) radial distribution function $g\left(r\right)$ vs radial distance $r$ in units of collapsed microgel outer radius $b_0$ and (b) static structure factor $S\left(q\right)$ vs scattered wave vector $q$ in solutions with system parameters of Fig. 15, collapsed inner radius $a_0=50$ nm, cross-link fraction $x=0.001$, and generalized volume fractions $\zeta$ from fluid (black long-dashed curve) to solid phase, which may be either an fcc crystal (red solid curve) or amorphous (blue short-dashed curve).

Figure 19

(a) SANS intensities, $I\left(q\right)$, vs scattering vector, $q$, of deuterated hollow 5 mol% cross-linked microgels below (blue circles) and above (red circles) the VPTT, at 20 and $40{}^{\circ }\mathrm{C}$, respectively. The solid lines represent the fit of the data with the model in Ref. [23]. (b) Radial distributions of the relative polymer volume fraction as obtained by the fits of the curves in panel (a) using the model of Ref. [23]. The sample concentration is $\zeta =0.081\pm 0.005$.

Figure 20

(a) SANS (circles) and SLS (triangles) scattered intensities $I\left(q\right)$ of the hydrogenated hollow 5 mol% cross-linked microgels vs scattering vector $q$. The solid curves represent the data fits using the model in Ref. [23]. The sample had a $\zeta =0.080\pm 0.003$ and a temperature $T=20.0\pm 0.01{}^{\circ }\mathrm{C}$. (b) Radial distribution of the relative polymer volume fraction within the microgel as obtained from the SANS and SLS data fit. Colors, symbols, and concentrations in panel (b) correspond to those in panel (a).

Figure 21

(a) Additional SANS intensities, $I\left(q\right)$, vs scattering vector, $q$, of hydrogenated hollow 5 mol% cross-linked microgels. (b) Radial distribution of the relative polymer volume fraction as obtained by fitting the curves in panel (a) using the model of Ref. [23]. In panel (a), the concentrations from bottom to top are $\zeta =0.20\pm 0.01$, $0.25\pm 0.01$, $0.35\pm 0.01$, $0.40\pm 0.01$, $0.45\pm 0.01$, $0.50\pm 0.01$, $0.55\pm 0.01$, $0.60\pm 0.01$, $0.70\pm 0.01$, $0.74\pm 0.01$, $0.76\pm 0.02$, $0.90\pm 0.02$, $0.95\pm 0.02$, $1.00\pm 0.02$, $1.10\pm 0.02$, and $\zeta <0.080\pm 0.003$. The top curve is at $T=40.0\pm 0.1{}^{\circ }\mathrm{C}$, while all the other measurements are at $T=20.0\pm 0.1{}^{\circ }\mathrm{C}$. In panel (b), the colors and therefore the concentrations correspond to the colors in panel (a).

Figure 22

Probability distributions of outer (a) and inner (b) swelling ratios of hollow microgels in solution with system parameters of Fig. 15, collapsed inner radius $a_0=32$ nm, cross-link fraction $x=0.001$, and generalized volume fraction $\zeta$ increasing from right to left as shown. Inset to panel (b) shows distributions for lowest $\zeta$ values on a larger scale.