Effects of linker flexibility on phase behavior and structure of linked colloidal gels

The phase behavior of a dilute mixture of colloids and semiflexible or rigid linkers is expected to be qualitatively similar to that of mixtures of patchy colloids [23] or colloids and flexible linkers [24]. The homogeneous colloid–linker mixture is typically thermodynamically stable when the attraction between the colloids and linkers is weak (small $\beta\varepsilon$) but can become unstable to composition fluctuations when the attraction is strong (large $\beta\varepsilon$) and enough linker is added (increasing $\Gamma$), leading to separation into colloid-rich and colloid-lean phases. We note that the stable fluid phase can be reentrant with respect to $\Gamma$ [23, 24], but our simulations are performed in the linker-limited regime where increasing $\Gamma$ destabilizes the fluid.

The thermodynamically unstable regions of the phase diagram are demarcated by the spinodal boundary [58, 61]. Mixtures prepared inside this region will spontaneously phase separate by spinodal decomposition, and this process typically produces kinetically arrested gels [20]. Mixtures prepared outside both the spinodal region and the binodal region (demarcating true phase coexistence) remain single phase but can form gels if the lifetime of the linker-mediated connections between colloids becomes long compared to the observation time [62, 20]. Because of their inherent thermodynamic stability, these “equilibrium gels” are more resistant to aging than gels formed by arrested spinodal decomposition. Theoretical knowledge of the phase diagram can guide design of materials that controllably gel by one these routes.

We previously applied Wertheim’s first-order thermodynamic perturbation theory (TPT) [63, 64, 65, 66, 67] to predict the conditions for phase separation in a mixture of colloids and fully flexible ($\beta\kappa=0$) linkers [24]. Within TPT, the Helmholtz free-energy density $a$ is decomposed into two contributions, $a=a_{0}+a_{\rm b}$. $a_{0}$ is the free-energy density of a reference system without bonding between components, and $a_{\rm b}$ is the free-energy density due to bonding. For fully flexible linkers, the hard-chain fluid [67] is a useful reference system and has

The first term is the free-energy density of an ideal gas of chain molecules,

where the sum is taken over the different components $i$, each of which consists of chains of $M_{i}$ beads of equal diameter $d_{i}$. Here, $\rho_{i}$ is the number density, and $\Lambda_{i}$ is the thermal wavelength of component $i$. For our model, there are two components: colloids each having $M_{\rm c}=1$ bead of diameter $d_{\rm c}$ and with $\rho_{\rm c}=6\eta_{\rm c}/(\pi d_{\rm c}^{3})$, and linkers having $M_{\rm l}=M$ beads of diameter $d$ and with $\rho_{\rm l}=\Gamma\rho_{\rm c}$. The second term in Eq. (7), $a_{\rm hs}^{\rm ex}$, is the excess free-energy density of the hard-sphere fluid obtained by dissociating all bonds from the chains [68, 69],

where $\xi_{j}=\sum_{i}\rho_{i}M_{i}\pi d_{i}^{j}/6$. The third term is the excess free-energy density of chain formation [67], where $g_{\rm hs}^{(ii)}(d_{i}^{+})$ is the contact value of the radial distribution function for two beads of component $i$ in the hard-sphere fluid [68]. Note that Eq. (7) corrects a typographical error in Eq. (5) of Ref. 24, where the ideal gas and excess hard-sphere free energies were incorrectly described; Ref. 24 in fact used Eq. (7) to perform calculations.

To compute the bonding free-energy density $a_{\rm b}$, it is usually assumed that to first order each site can bond with at most one other site, that bonding is uncorrelated between sites, and that only treelike bonded networks form [70]. For a mixture of components each having $n_{i}$ identical bonding sites, $a_{\rm b}$ can be expressed as [67]

where $X_{i}$ is the fraction of sites not bonded on component $i$. In our model, the colloids have $n_{\rm c}=6$ sites and the linkers have $n_{\rm l}=2$ sites. The values of $X_{i}$ are determined by the chemical equilibrium equations,

where only $\Delta_{\rm cl}$ is nonzero for our model and is given by [71]

Here, $\mathbf{r}$ is the vector from the center of the colloid to the end of the linker participating in the bond, $\boldsymbol{\Omega}$ is the vector of Euler angles defining the orientation of the colloid, and $\mathbf{R}$ is the end-to-end vector of the linker. The integrand is the product of the distribution of end-to-end vectors $p_{0}(\mathbf{R})$ in the hard-chain reference fluid, the colloid–linker pair correlation function $g_{0}^{({\rm cl})}$ in the hard-chain reference fluid, and the Mayer $f$-function $f^{({\rm cl})}$ for a designated bonding site on the colloid and the end of the linker at $\mathbf{r}$. The $f$-function, and hence the integrand, is nonzero only when the two sites interact, which occurs over a short range given the form of Eq. (4). $\Delta_{\rm cl}$ can be regarded as a “bond volume” averaged over orientations of the colloid and conformations of the linker. We evaluated $\Delta_{\rm cl}$ using the same methodology as in Ref. 71 [72, 73, 74], where $p_{0}$ and $g_{0}^{({\rm cl})}$ were obtained from simulations of the hard-chain reference fluid; complete details are given in the Supporting Information.

Given the Helmholtz free energy $a$, the spinodal boundary can be computed using the matrix $\mathbf{H}$ of second derivatives with respect to density [61], where $H_{ij}=\partial^{2}a/\partial\rho_{i}\partial\rho_{j}$. The limit of stability occurs when the determinant of $\mathbf{H}$ is zero. For the fully flexible linkers ($\beta\kappa=0$), the TPT predictions are in good agreement with the simulations (Figure 2a). (The TPT predictions are also improved compared to Ref. 24 because Eq. (12) uses a more accurate approximation of the pair correlation function.) Phase separation was mainly observed in the simulations inside the TPT spinodal; Figure 2b shows a representative phase-separated morphology at $\eta_{\rm c}=0.03$ and $\beta\varepsilon=20$. We previously investigated how the phase behavior of mixtures of colloids and fully flexible linkers depended on the linker length [24], so here we focus on the role of flexibility.

As a perturbation theory, the TPT free energy is sensitive to the reference fluid model, i.e., without linking. In particular, the free-energy density $a$ depends on both the reference fluid’s equation of state, which determines $a_{0}$, and its structure, which enters $a_{\rm b}$ through $\Delta_{\rm cl}$. We hence expect similar phase behavior for different bending stiffnesses $\kappa$ if the equation of state and relevant structures of the reference fluid do not depend strongly on $\kappa$. We performed simulations of mixtures having $\beta\varepsilon=0$, $\Gamma=1.5$, and varied $\eta_{\rm c}$ and $\kappa$, and we measured the pressure $P$ (Figure 2d), the distribution of linker end-to-end vectors $p_{0}(\mathbf{R})$ (Figure 3), and the pair distribution function $g_{0}^{({\rm cl})}$ (Figures S1 and S2). The pressure changed negligibly with $\kappa$ and was fully consistent with the pressure derived from Eq. (7) [67]. The end-to-end vector distribution $p_{0}(\mathbf{R})$ changed significantly with $\kappa$, shifting its peak to larger $R=|\mathbf{R}|$ for larger $\kappa$ as expected. The pair distribution function $g_{0}^{({\rm cl})}$ was again far less sensitive to $\kappa$. It tended to be less than 1 for conformations relevant to evaluating Eq. (12) due to depletion of the polymer-like chains near the surface of the colloid [75], with slightly larger values for the stiffer linkers. Overall, though, the change in $\Delta_{\rm cl}$ at $\beta\varepsilon=20$ was only 30% between $\beta\kappa=0$ and $\beta\kappa=28$.

Given this analysis of the reference fluid, we expected similar phase behavior at all values of $\kappa$. Indeed, for $\beta\kappa\gtrsim 10.5$ (the rigid linkers), the simulated phase diagrams were essentially the same, closely resembling that at $\beta\kappa=0$ and the TPT predictions. Surprisingly, though, the semiflexible linkers ($1.75\leq\beta\kappa\lesssim 7$) exhibited very different behavior than predicted by first-order TPT. For $\beta\kappa=1.75$ and $3.5$, only minor structuring was detected at all $\eta_{\rm c}$ and $\varepsilon$, with $S_{\rm cc}(0)$ well below the threshold we considered phase separated. This difference was visually apparent in simulation snapshots taken at conditions inside the TPT spinodal. Unlike when $\beta\kappa=0$ (Figure 2b), significant colloid density variations were not visible at $\eta_{\rm c}=0.03$ and $\beta\varepsilon=20$ when $\beta\kappa=3.5$ (Figure 2c). The lack of phase separation for only the semiflexible linkers could not be explained using first-order TPT.

We suspected that the striking discrepancy between the simulations and TPT for the semiflexible linkers was caused by certain linking motifs that are prevalent in the simulations at these flexibilities but are not accounted for in first-order TPT. In previous work, we have speculated that the presence of cycles within the graph representing the bonded network of colloids and linkers might suppress phase separation [24, 22]. In particular, we hypothesized that for a fixed total amount of linker $\Gamma$ the fluid phase might be stabilized if a substantial number of linkers formed “loops” by attaching both ends to the same colloid, rather than forming “bridges” between two different colloids; a similar explanation was recently proposed for the lack of gelation in DNA hydrogels linked with flexible joints [26]. This definition of a loop, which can also be interpreted as a “double bond” between a linker and a colloid, is the smallest cycle that can form, and other cycles like double links between two colloids are included with the bridges here.

We counted the number of linkers forming either bridges or loops in our simulations (Figure 4), focusing on the strong attraction regime ($\beta\varepsilon=20$) where essentially all linkers had both ends bonded in one of these motifs. For the fully flexible linkers, a small fraction of linkers formed loops (Figure 4b) rather than bridges (Figure 4a) at all $\eta_{\rm c}$, with the loop fraction tending to be smaller at larger $\eta_{\rm c}$. An overwhelming majority of rigid linkers formed bridges at all $\eta_{\rm c}$, constituting $\geq 90\%$ of the links for $\beta\kappa\geq 14$. Interestingly, and consistent with our hypothesis, there was a pronounced increase in the loop fraction (Figure 4b) for the semiflexible linkers, which had a maximum at roughly $\beta\kappa=3.5$. At $\eta_{\rm c}=0.03$, nearly 50% of the linkers formed loops when $\beta\kappa=3.5$ and the mixture remained essentially single phase, despite phase separating for both the fully flexible and rigid linkers (Figure 2a).

We recently extended first-order TPT to include linker loops (double bonds) [71], and this theory readily predicts the fraction of linkers forming bridges or loops in a spatially homogeneous fluid at equilibrium. We computed these fractions at $\eta_{\rm c}=0.03$ and $\eta_{\rm c}=0.15$ for all chain flexibilities using a similar methodology as in Ref. 71; numerical details are given in the Supplementary Material. The TPT predictions for the fraction of linkers forming loops are in excellent agreement with the simulations (Figure 4), supporting the notions that (1) loops are equilibrium bonding motifs and (2) for the studied colloid bonding site geometry and linker length, loops are more prevalent at intermediate flexibilities. We note that the TPT neglects any phase separation in the simulations at $\eta_{\rm c}=0.03$; this does not seem to qualitatively affect the result.

The maximum in loop fraction with respect to $\kappa$ can be understood using the distribution of linker end-to-end vectors (Figure 3). In order to form a loop, both ends of the linker must attach to sites on the same colloid, so loops become more prevalent when it is more probable for the linker to have an end-to-end distance commensurate with the colloid site–site distance. We consider only loops between patches at neighboring vertices of the octahedron, which are separated by distance $\sqrt{2}d_{\rm cl}^{*}$ (arrow in Figure 3), because loops between vertices on opposite hemispheres of the colloid are forbidden by the linker contour length. As $\kappa$ increases, this end-to-end distance initially becomes more probable but then becomes significantly less likely once the linkers become rigid. The average end-to-end distance $\langle R\rangle$ (inset of Figure 3) is also comparable to the site–site distance (dotted line) in the semiflexible regime where the most loops are observed. Qualitatively, the flexible linkers pay an entropic penalty to stretch beyond their preferred size and form a loop, while the rigid linkers pay a bending-energy penalty to do the same; in contrast, the semiflexible linkers are naturally compatible with the length scale required to form loops in our model. These effects of flexibility measured by $p_{0}$ manifest in the TPT as an increase in the double-bond volume [71], which is nearly three times larger when $\beta\kappa=3.5$ than when $\beta\kappa=0$.

An immediate consequence of this interpretation is that the loop fraction should also depend on the distribution of distances $\xi$ between bonding sites on the colloids. A distribution $p(\xi)$ that overlaps significantly with $p_{0}(\mathbf{R})$ should favor loop formation, while an incompatible $p(\xi)$ should produce fewer loops. The impact of the colloid bonding site geometry on loop formation is not only of theoretical interest but also a practical consideration for experiments, where $p(\xi)$ might be determined by the details of the surface functionalization and polydispersity in surface characteristics is inherent to synthesis. To test the sensitivity of properties to the bonding site geometry, we randomly displaced the colloid bonding sites from their initial positions at the vertices of an octahedron. As illustrated in Figure 5, each site was pertubed by a uniformly random amount on the surface of the sphere with radius $d_{\rm cl}^{*}$ up to a maximum polar angle $\phi^{*}$. Adjusting $\phi^{*}$ varies $p(\xi)$ from a perfectly monodisperse distribution at $\phi^{*}=0$ to a broad distribution at $\phi^{*}=0.6$ (Figure 5), which is close to the maximum polar angle that still guarantees two sites will not overlap. (We include in $p(\xi)$ only the distances between sites that were initially nearest neighbors on the octahedron, as these are the ones most likely to form loops even after being displaced.) Generating bonding site polydispersity in this way ensures that the average site–site distance $\langle\xi\rangle$ remains nearly constant (decreasing only 2% over the entire $\phi^{*}$ range) and allows us to isolate the effects of site polydispersity from effects of the average site–site distance.

We simulated the colloid–linker mixture with polydisperse bonding sites at $\eta_{\rm c}=0.03$, which is inside the first-order TPT spinodal and led to phase separation for the flexible and rigid linkers. We will again focus on the loop fraction and phase behavior in the strong attraction limit $\beta\varepsilon=20$. For $\phi^{*}\leq 0.2$, the bridge and loop fractions were nearly unchanged from the monodisperse ($\phi^{*}=0$) case, with the loop fraction having a maximum for the semiflexible linkers (Figure 6). Further increase in $\phi^{*}$ significantly increased the loop fraction for the flexible linkers, but the rigid linkers were less affected. It is sensible that increasing $\phi^{*}$ had a more dramatic effect on the flexible linkers than the rigid linkers because $p_{0}(\mathbf{R})$ for the flexible linkers is broad (Figure 3); perturbing the colloid bonding sites brings some sites closer together (Figure 5) where there are probable flexible-linker conformations that previously did not produce loops. For the rigid linkers, increasing $\phi^{*}$ did little to increase the overlap between the two distributions, and the loop fraction correspondingly changed little with $\phi^{*}$.

We also measured the colloid partial structure factor $S_{\rm cc}(0)$ (Figure 7a) to interrogate how the thermodynamic stability of the fluid phase depended on $\phi^{*}$. We show only the results for $\beta\kappa\leq 7$ because all rigid-linker mixtures phase separated regardless of $\phi^{*}$. Based on the measured loop fractions, we expected and observed similar phase behavior when $\phi^{*}\leq 0.2$ as for the monodisperse ($\phi^{*}=0$) bonding site geometry, with $S_{\rm cc}(0)$ consistent with a lack of phase separation in the semiflexible-linker mixtures. For larger $\phi^{*}$, phase separation was also suppressed for the flexible linkers, which had a significant increase in the loop fraction. Indeed, we found a strong negative correlation between $S_{\rm cc}(0)$ and the loop fraction (Figure 7b); $S_{\rm cc}(0)$ indicated phase separation when less than 30% of linkers formed loops. This strongly suggests that controlling the loop fraction, e.g., by designing the linker length or flexibility, is a key component of engineering the phase behavior of the colloid–linker mixture and the conditions that may lead to gelation.

So far, we have focused our discussion on controlling phase behavior, but in our previous work with flexible linkers, we showed that adjusting the linker length also gave control over the microstructure of the assembled colloids [24]. Longer linkers having larger $\langle R\rangle$ tended to increase the spacing between colloids, which may have important consequences for the optical properties of gels containing plasmonic nanocrystals [4, 5]. Linker flexibility gives another potential handle to control the colloid microstructure because semiflexible or rigid linkers have larger $\langle R\rangle$ than flexible linkers of equal contour length (Figure 3).

We measured the colloid–colloid pair correlation function $g^{({\rm cc})}$ (Figure S3) for various compositions and flexibilities in the strong attraction limit ($\beta\varepsilon=20$). We focused on the position $r_{\rm max}$ of the maximum in $g^{({\rm cc})}$ (Figure 8a), which identifies the typical distance to a colloid’s nearest neighbor. At small $\eta_{\rm c}$, $r_{\rm max}$ shifted to larger distances with increasing $\kappa$, which is consistent with the corresponding increase in $\langle R\rangle$. Interestingly, this shift did not depend on whether the mixture phase separated, and $r_{\rm max}$ was nearly constant for a given $\kappa$ when $\eta_{\rm c}\lesssim 0.10$. At larger $\eta_{\rm c}$, $r_{\rm max}$ decreased with increasing $\eta_{c}$ and ultimately reached nearly the same value for all $\kappa$ at $\eta_{c}=0.15$. The flexible linker seems to be an exception, having nearly constant $r_{\rm max}$ over the entire range of $\eta_{c}$ we simulated.

Due to many-body effects in the linked-colloid assemblies, the measured $r_{\rm max}$ cannot be simply explained using only the average end-to-end size of the linker $\langle R\rangle$. Assuming a colinear arrangement of two colloids and a linker, we might expect $r_{\rm max}\approx 2d_{\rm cl}^{*}+\langle R\rangle$, which is $9.7\,d$ and $12.7\,d$ for $\beta\kappa=0$ and $28$, respectively; both are considerably larger than the measured $r_{\rm max}$. To obtain a smaller $r_{\rm max}$, the linkers must either compress or connect two colloids at an angle with each other. A significant decrease in the end-to-end distance of the rigid linkers would incur a large bending penalty and is unlikely, so we accordingly measured the bond angle $\theta$ between pairs of linked colloids (Figure 8b). We defined $\theta$ using the center and bonded patch of the first colloid and the bonded patch on the second colloid. The distribution of bond angles showed predominantly colinear arrangements ($\theta=\pi$) for the flexible linkers, but as $\kappa$ increased, the bond angles were forced toward more right-angle ($\theta=\pi/2$) arrangements. This change in $\theta$ allows the colloids to pack closely without compressing the linker and seems to supress formation of a single dominant length scale in the assemblies, producing a more uniform distribution of pair distances (Figure S3).