Gelation in mixtures of polymers and bidisperse colloids

We investigated the effects of varying the volume fraction of large particles $\left(r\right)$ on the linear rheology and microstructure of mixtures of polymers and bidisperse colloids, in which the ratio of the small and large particle diameters was $\alpha =0.31$ or $\alpha =0.45$. Suspensions formulated at a total volume fraction of ${\varphi }_T=0.15$ and a constant concentration of polymer in the free volume $c/c^{*}\approx 0.7$ contained solid-like gels for small $r$ and fluids or fluids of clusters at large $r$. The solid-like rheology and microstructure of these suspensions changed little with $r$ when $r$ was small, and fluidized only when $r>0.8$. By contrast, dense suspensions with ${\varphi }_T=0.40$ and $\alpha =0.31$ contained solid-like gels at all concentrations of large particles and exhibited only modest rheological and microstructural changes upon varying the volume fraction of large particles. These results suggest that the effect of particle-size dispersity on the properties of colloid-polymer mixtures are asymmetric in particle size and are most pronounced near a gelation boundary.

Figure 1

(a)–(f) Storage ($G^{\prime }$, closed symbols) and loss ($G^{\prime \prime }$, open symbols) moduli as a function of frequency (stress amplitude = $2\times {10}^{-4}$ Pa) for suspensions with total volume fraction ${\varphi }_T\approx 0.15$, polymer concentration $c_p=25$ mg/ml, and particle size ratio $\alpha \approx 0.31$. Volume fraction of large particles $r$: (a) 0.00, (b) 0.50, (c) 0.75, (d) 0.87, (e) 0.96, (f) 1.00. (g) Elastic modulus at a frequency of ${10}^{-2}$ Hz, $G^{\prime }(\omega ={10}^{-2}$ Hz), as a function of $r$.

Figure 2

Normalized mean-squared displacement MSD/$\left(2a_L\right){}^2$ as a function of lag time $\tau$ for large particles in suspensions with total colloid volume fraction ${\varphi }_T\approx 0.15$, polymer concentration $c_p=25$ mg/ml, and particle size ratio $\alpha \approx 0.31$ but varying volume fraction of large particles $r$. The dashed lines indicate power-law fits (MSD $\sim {\tau }^{\beta }$), with the exponents $\beta$ reported next to each fit.

Figure 3

(a)–(f) Confocal micrographs of small and large particle populations for suspensions with total volume fraction ${\varphi }_T\approx 0.15$, polymer concentration $c_p=25$ mg/ml, particle size ratio $\alpha \approx 0.31$, and volume fraction of large particles $r$ of (a) 0.00, (b) 0.50, (c) 0.75, (d) 0.87, (e) 0.93, and (f) 1.00. The scale bar is $10 \mu \mathrm{m}$. (g) Probability distribution of the number of large-large particle bonds. Inset: Average bond number $n_{\mathrm{avg}}$ as a function of $r$.

Figure 4

Summary of mechanical, dynamic, and microstructural characterizations for suspensions with ${\varphi }_T\approx 0.15$, polymer concentration $c_p=25$ mg/ml, and particle size ratio $\alpha \approx 0.45$. (a) Elastic modulus at a fixed frequency $\omega ={10}^{-2}$ Hz as a function of volume fraction of large particles $r$. (b) Normalized mean-squared displacement MSD/$\left(2a_L\right){}^2$ as a function of lag time $\tau$ for large particles. The dashed lines indicate power-law fits (MSD $\sim {\tau }^{\beta }$), with the exponents $\beta$ reported next to each fit. (c) Average bond number $n_{\mathrm{avg}}$ as a function of $r$.

Figure 5

(a)–(f) Storage ($G^{\prime }$, closed symbols) and loss ($G^{\prime \prime }$, open symbols) moduli as a function of frequency (stress frequency = $1\times {10}^{-3}$ Pa) for suspensions with total volume fraction ${\varphi }_T\approx 0.40$, polymer concentration $c_p=25$ mg/ml, and particle size ratio $\alpha \approx 0.31$. Volume fraction of large particles $r$: (a) 0.00, (b) 0.50, (c) 0.75, (d) 0.87, (e) 0.93, (f) 1.00. (g) Elastic modulus at a frequency of ${10}^{-2}$ Hz, $G^{\prime }(\omega ={10}^{-2}$ Hz), as a function of $r$.

Figure 6

Normalized mean-squared displacement MSD/$\left(2a_L\right){}^2$ as a function of lag time $\tau$ for large particles in suspensions with total colloid volume fraction ${\varphi }_T\approx 0.40$, concentration of polymer $c_p=25$ mg/ml, and particle-size ratio $\alpha \approx 0.31$ but varying volume fraction of large particles $r$. The dashed lines indicate power-law fits (MSD $\sim {\tau }^{\beta }$). Typical exponents $\beta$ are reported near representative fit lines.

Figure 7

(a)–(f) Confocal micrographs of small- and large-particle populations for suspensions with total volume fraction ${\varphi }_T\approx 0.40$, polymer concentration $c_p=25$ mg/ml, particle-size ratio $\alpha =a_S/a_L=0.31$, and varying volume fraction of large particles $r$ of (a) 0.00, (b) 0.50, (c) 0.75, (d) 0.87, (e) 0.93, and (f) 1.00. The scale bar is $10 \mu \mathrm{m}$. (g) Probability distribution of the number of large-large particle bonds. Inset: Average bond number $n_{\mathrm{avg}}$ as a function of $r$.

Figure 8

Summary of metrics for mechanical and structural changes with dispersity ratio. (a) Elastic modulus at a frequency of ${10}^{-2}$ Hz, $G^{\prime }(\omega ={10}^{-2}$ Hz) and (b) average bond number $n_{\mathrm{avg}}$ as a function of volume fraction of large particles $r$ for all three series of experiments: Series 1 ($\varphi =0.15$ and $\alpha =0.31$, circles), Series 2 ($\varphi =0.15$ and $\alpha =0.45$, triangles), and Series 3 ($\varphi =0.40$ and $\alpha =0.31$, squares).

Figure 9

Normalized deviation of the van Hove correlation function from the Gaussian function for single-particle displacements, $(G_S(\mathrm{\Delta }x,\tau )-G_{S,G}(\mathrm{\Delta }x,\tau ))/G_{S,G}(\mathrm{\Delta }x,\tau )$ at a lag time $\tau =50$ sec for (a) ${\varphi }_T\approx 0.15,\alpha \approx 0.31$; (b) ${\varphi }_T\approx 0.15,\alpha \approx 0.45$; (c) ${\varphi }_T\approx 0.40,\alpha \approx 0.31$. Inset: localization length $x_{\mathrm{loc}}$ extracted from the first zero of $(G_S(\mathrm{\Delta }x,\tau )-G_{S,G}(\mathrm{\Delta }x,\tau ))/G_{S,G}(\mathrm{\Delta }x,\tau )$ as a function of the volume fraction of large particles $r$.

Figure 10

Dimensionless electrostatic (dotted lines), depletion (dashed lines), and overall (solid lines) between spherical particles as a function of the normalized interparticle distance $r/2a_L$ (for large particles, (a), (c), (e)) or $r/2a_S$ (for small particles, (b), (d), (f)) for polymer concentration $c_p=25$ mg/ml. (a), (b): Series 1 experiments: $\varphi =0.15$ and $\alpha =a_S/a_L=0.31$ ($2a_L=1.76 \mu \mathrm{m}$ or $2a_S=0.54 \mu \mathrm{m}$); (c), (d): Series 2 experiments: $\varphi =0.15$ and $\alpha =a_S/a_L=0.45$ ($2a_L=1.57 \mu \mathrm{m}$ or $2a_S=0.71 \mu \mathrm{m}$); (e), (f): Series 3 experiments: $\varphi =0.40$ and $\alpha =a_S/a_L=0.31$ ($2a_L=1.76 \mu \mathrm{m}$ or $2a_S=0.54 \mu \mathrm{m}$).