Open Access Article

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Estefania
Vidal-Henriquez
* and
David
Zwicker
*

Max-Planck Institute for Dynamics and Self-Organization, Am Faβberg 17, 37077 Göttingen, Germany. E-mail: estefania.vidal@ds.mpg.de; david.zwicker@ds.mpg.de

Received
31st January 2020
, Accepted 2nd June 2020

First published on 8th June 2020

Liquid droplets embedded in soft solids are a new composite material whose properties are not very well explored. In particular, it is unclear how the elastic properties of the matrix affect the dynamics of the droplets. Here, we study theoretically how stiffness gradients influence droplet growth and arrangement. We show that stiffness gradients imply concentration gradients in the dilute phase, which transport droplet material from stiff to soft regions. Consequently, droplets dissolve in the stiff region, creating a dissolution front. Using a mean-field theory, we predict that the front emerges where the curvature of the elasticity profile is large and that it propagates diffusively. This elastic ripening can occur at much higher rates than classical Ostwald ripening, thus driving the dynamics. Our work shows how gradients in elastic properties control the arrangement of droplets, which has potential applications in soft matter physics and biological cells.

Physical experiments with synthetic materials can help to understand how an elastic environment affects droplet growth. For instance, oil droplets growing in a homogeneous PDMS gel form mono-disperse emulsions and the stiffness of the gel controls droplet size.^{9} There are several advantages of these experiments: first, the system is not driven, implying it relaxes toward equilibrium after preparation. Second, the gel is strongly cross-linked, so it does not spontaneously rearrange on the time scale of the experiments.^{10} Viscous relaxation is thus negligible. Third, droplets are large compared to the mesh size, implying that the gel can be described by a continuum theory. Taken together, these properties allow to isolate the effects of an elastic environment on droplets.

The aim of the present paper is to understand how spatially varying stiffnesses affect droplet dynamics. This is motivated by recent experiments showing that stiffness gradients lead to elastic ripening, where droplets dissolve in stiffer regions.^{11} Moreover, these experiments revealed a dissolution front invading stiffer regions, while the material of the dissolving droplets accumulated in softer regions. We already developed a theoretical description of the situation, which is based on the assumption that the gel exerts a pressure onto droplets that is proportional to the local stiffness.^{11} Numerical simulations of this theory showed excellent agreement with the measured data. In the present paper, we analyze this model numerically and analytically to understand the details of elastic ripening. In particular, we use a simplified, coarse-grained version to derive scaling laws for where and when the dissolution front starts and how it evolves in time. These simple equations allow us to predict how the model parameters affect elastic ripening in more complex situations, e.g., when the elasticity profile varies in all spatial dimensions.

In the case of a single spherical droplet of radius R embedded in an isotropic elastic matrix, the droplet exhibits an additional pressure P due to both surface tension and elastic effects. In the simple case of a dense droplet phase and an ideal dilute phase, the resulting equilibrium volume fraction in the dilute phase can be approximated as

(1) |

The pressure P on a single spherical droplet of radius R embedded in an elastic gel is

(2) |

To see whether surface tension γ is important in the ripening experiments, we next estimate the relevant pressure differences between droplets. The pressure difference generated by surface tension is roughly γΔR/R^{2}, where R is the mean droplet radius and ΔR denotes the difference in the radii. Conversely, the pressure difference created by the external gel can be estimated by the typical stiffness difference ΔE. Surface tension is thus negligible if the ratio

(3) |

(4) |

(5) |

(6) |

The dynamics of the system is governed by two diffusive fluxes that act on different length scales. Locally, material is exchanged between the droplets and the dilute phase by the flux J_{i}. Conversely, transport on longer length scales only happens in the dilute phase by the redistribution flux −D∇ϕ.

(7) |

Fig. 1 shows the time course of a typical simulation. Starting from the homogeneous initial condition, the system quickly forms two separate regions aligned with the stiffness profile (upper panel). Here, the stiff side exhibits smaller droplets and larger volume fractions in the dilute phase compared to the soft side. Droplets then start dissolving in the transition region and a dissolution front moves into the stiff side. Simultaneously, droplets grow on the soft side of the transition region while droplets far into the soft side remain unchanged. The exact same dynamics have been observed in the elastic ripening experiments;^{11} see Fig. 2. These dynamics can be understood qualitatively by considering the diffusive fluxes in the system.

Fig. 1 Numerical simulation showing a dissolution front invading the stiff region defined by a sigmoidal elasticity profile (upper panel). Subsequent images show projections of 3-dimensional simulations (obtained by solving eqn (5) and (6) of the droplets (symbols) and the volume fraction ϕ in the dilute phase (density plot with color bar at the bottom) at the indicated times. The model parameters are ϕ_{0} = 0.033, ϕ_{in} = 1, E_{stiff} = 0.15Ê, E_{soft} = 10^{−4}Ê, and w = 1.45. Here, = (V_{sys}/N_{drop})^{1/3} is a typical droplet separation with associated diffusive time t_{D} = ^{2}/D, where V_{sys} is the system's volume and N_{drop} is the total number of droplets. The mean droplet radius on the stiff side is 0.08. |

Fig. 2 Our model quantitatively captures the dynamics of dissolution fronts in the elastic ripening experiments.^{11} Shown are the experimental and numerical mean droplet radii as a function of the distance from the interface on the stiff side for five different time points. The model parameters are E_{stiff} = 0.0341Ê, E_{soft} = 3.2 × 10^{−4}Ê, w = 0.37, and the mean droplet radius on the stiff side is 0.09; see ESI,† for further details on the comparison. |

In the initial stage, the system is supersaturated everywhere, ϕ > ϕ_{eq}. Consequently, material is transferred from the dilute phase to the droplets until a local equilibrium is reached, ϕ = ϕ_{eq}. Eqn (4) implies that ϕ_{eq} is smaller for softer regions, so more material is absorbed by the droplets. We thus observe larger droplets in softer regions (see Fig. 1), consistent with experimental observations.^{9}

After the initial, local equilibration, material redistribution on longer length scales sets in. Since the stiffer side exhibits larger volume fractions ϕ in the dilute phase, material is transported to the soft side. Consequently, on the stiff side, ϕ drops below the local equilibrium volume fraction ϕ_{eq}, droplets shrink, and eventually dissolve. This process starts close to the transition region, since the redistribution flux is driven by gradients in ϕ, which do not exist further away. Once droplets start disappearing in the transition region, droplets further away begin to be affected and a dissolution front forms that invades the stiff side. All the material redistributed from the stiff side is absorbed by the droplets on the soft side close to the transition region, which effectively shield all the other droplets on the soft side.

(8) |

The dynamics of the coarse-grained system follow from eqn (5) and (6). We show in the ESI,† that the dynamical equations are

(9a) |

∂_{t}ϕ = D∇^{2}ϕ − ∂_{t}ψ. | (9b) |

Fig. 3 shows that the results of the numerical simulation of the coarse-grained model are virtually indistinguishable from that of the detailed model. Therefore, the coarse-grained model captures the essential features of the elastic ripening process. In particular, the dynamics of the dissolution front are governed by the material distribution, while individual droplets are irrelevant.

Fig. 3 The coarse-grained model (red lines, eqn (9)) captures the detailed dynamics of the full model (blue lines, eqn (5) and (6)). Shown are the profiles of the volume fractions ϕ in the dilute phase (upper panels) and the fraction ψ contained in droplets (lower panels) for three different time points t. The model parameters are given in Fig. 1. |

After the initial equilibration stage, the inhomogeneities in E, and thus in ϕ, drive diffusive fluxes toward the soft side. However, we observe that these fluxes mostly affect ψ and hardly change ϕ before the first droplets disappear. To understand the dynamics in this stage, we approximate eqn (9b) by ∂_{t}ψ ≈ D∇^{2}ϕ_{eq}. Consequently, ψ evolves as

ψ(,t) ≈ ψ_{0}() + tD∇^{2}ϕ_{eq} | (10) |

We can use eqn (10) to estimate the time and position of the start of the dissolution front. In particular, droplets dissolve after a time τ_{*}() ≈ −ψ_{0}()/(D∇^{2}ϕ_{eq}), when all material is removed from the droplet phase. The dissolution front starts at the earliest of these time points, τ_{start} = min_{}(τ_{*}|τ_{*} ≥ 0), which is given by

(11) |

The starting time τ_{start} can be estimated for the simple stiffness profile given by eqn (7). In particular, the droplet volume fraction ψ_{0} will be close to the value ψ_{stiff} deep into the stiff side; the curvature is approximately ∇^{2}ϕ_{eq} ∼ Δϕ/w^{2}, where Δϕ = ϕ_{eq}(E_{stiff}) − ϕ_{eq}(E_{soft}) denotes the difference in the equilibrium volume fractions between the two sides and w is the width of the transition region. Using these estimates, eqn (11) suggests a time scale

(12) |

We test the prediction of eqn (11) and the scaling discussed above by comparing to numerical simulations of the full model; see Fig. 4. The collapse of the starting times shown in the left panel indicates that is the relevant time scale for this process. Moreover, the actual prediction τ_{start} given in eqn (11) is within a factor of two of the measured data. This analysis shows that the front appears earlier for larger stiffness differences (larger Δϕ) between the two sides, for narrower transitions regions (smaller w), as well as when there is less material in the droplet phase (small ψ_{stiff}). Fig. 4b shows the corresponding starting positions, which clearly are determined by the width w of the transition region. The data collapse indicates that neither the absolute stiffnesses nor the droplet size affects where the front appears.

Fig. 4 The starting time τ_{start} and position s_{start} of the front scale with the predicted time and length scales, respectively. Results from numerical simulations of the full model (dots) for various parameters are compared to the theoretical predictions from the coarse-grained model (gray line). The time point τ_{start} of the first dissolving droplet is shown in panel (a) as a function of the predicted associated time scale given in eqn (12). The theoretical prediction given by eqn (11) is shown for ψ = 0.09ϕ_{0} and w = 1.45. Panel (b) shows the associated starting position s_{start} together with the equivalent prediction following from eqn (11), which is shown for ψ = 0.09ϕ_{0} and E_{stiff} = 0.15Ê. The remaining parameters are E_{soft} = 10^{−4}Ê and ϕ_{in} = 1. |

(13) |

(14) |

(15) |

We test the theoretical prediction given in eqn (15) by comparing to numerical simulations of the full model. Fig. 5 shows the recorded times and positions when droplets dissolved (gray symbols), thus marking the dissolution fronts. The fronts start in the transition region on the stiff side and then move in opposite directions. The front moving toward the soft side slows down and comes to a halt on the soft side of the transition region, as predicted in the previous section. Conversely, the front invading the stiff side is quicker and does not stop. We measure its speed by fitting eqn (15) to the front positions deep into the stiff side to extract α and t_{0}; see Fig. 5a. Since the model explains the measured data at late times, we conclude that the front moves diffusively.

Fig. 5 Dissolution fronts move diffusively. (a) Position and time points of dissolving droplets in a simulation of the full model (dots) are compared to a fit (red line) of the theoretical prediction given in eqn (15). Model parameters are E_{stiff} = 0.09Ê and ψ = 0.09ϕ_{0}. Remaining parameters are given in Fig. 1. (b) The front diffusivity α (dots) determined from fitting to numerical simulations is compared to the prediction (line) given by eqn (14). Simulations were done for ψ_{stiff}/ϕ_{0} = 0.03, 0.09, 0.30 for various E_{stiff}, while the remaining parameters are the same as in panel (a). |

The fitted front diffusivity α is presented in Fig. 5b as a function of the relevant non-dimensional parameter Δϕ/ψ_{stiff}. This parameter compares the strength Δϕ of the elastic ripening to the fraction ψ_{stiff} of material that needs to be removed from the droplets. Consequently, the front is faster for larger Δϕ/ψ_{stiff}. The theoretical prediction for α, given in eqn (14), matches the data well for α < 2D. The fact that the front diffusivity α needs to be smaller than or comparable to the molecular diffusivity D is not surprising since we assumed that the front is slow enough for the region devoid of droplets to be in a stationary state; see eqn (13). Consequently, our theory predicts a maximal front diffusivity of 4D while the simulations indicate that faster fronts are possible.

Fig. 6 shows a numerical simulation of the full model for a two-dimensional elasticity profile (left panel). A detailed simulation of a similar pattern has already identified that droplets accumulate in the soft valleys^{8} and the time course shown in the right panels of Fig. 6 confirms that droplets follow the dynamics described above. A movie of this simulation, as well as one for a more complex elasticity profile, can be found in the ESI.† Taken together, this shows that we can engineer elasticity profiles to locate droplets in precise arrangements.

Elastic ripening competes with Ostwald ripening since both effects are driven by pressure differences between droplets. Their relative importance is quantified by , given by eqn (3), when ΔE is the relevant stiffness difference in the droplets' surroundings. For instance, although elastic ripening dominates initially in the simulation shown in Fig. 6, Ostwald ripening will eventually occur between the remaining islands because they have the same elastic properties (ΔE ≈ 0). A similar situation occurs in heterogeneous elastic materials, where ΔE corresponds to local stiffness variations. Thus, both elastic ripening and Ostwald ripening can happen in the same system, on different length- and time-scales.

The elastic ripening in stiffness gradients is similar to other droplet coarsening dynamics in gradient systems. For instance, concentration gradients, e.g., of regulating species that compete for mRNA binding,^{16} have been shown to bias droplet locations in experimental^{1} and theoretical studies.^{17–19} Similarly, other external fields, like temperature gradients created by local heating^{20,21} or even gravity^{22} could be used to control droplet arrangements. Such systems can be analyzed using approaches that are similar to the ones presented here.

We showed that elastic ripening allows to control droplet arrangements, which could for instance be used in technical applications for micropatterning or for creating structural color. Moreover, our theory can help to understand the localization of biomolecular condensates in biological cells. For instance, elastic ripening explains experiments where droplets have been induced in the stiff regions of heterochromatin, but moved into softer regions immediately.^{8} We expect that similar processes happen in the cytosol, where biomolecular condensates should be less likely where the cytoskeleton is dense. Interestingly, there are counterexamples, like centrosomes that localize to regions of high microtubule density^{23–25} or ZO-1 clusters that concentrate in the acto-myosin cortex.^{26,27} This seems to contradict elastic ripening, but in both examples the condensates interact with the elastic matrix: centrosomes bind the tubulin of microtubules^{28} and the ZO-1 protein interacts with the F-actin of the cortex.^{29} Consequently, there are two competing gradients in this situation: droplets are repelled by the stiffness of the surrounding matrix but are attracted by its molecular components. Indeed, when the actin-binding domain of ZO-1 is removed, the clusters do not accumulate in the cortex anymore, but are more broadly distributed,^{26} as predicted by elastic ripening.

Our theoretical description of elastic ripening can be naturally extended to include other effects. In fact, the dynamics described by eqn (5) and (6) already include surface tension effects when the pressure given by eqn (2) is used. Although we did not analyze the impact of surface tension since it is negligible in the experiments, it will become important on longer timescales or when surface tensions are large. Moreover, the elastic properties of the matrix might be more complex than considered here. For instance, the cytoskeleton can shown strain-stiffening, which might arrest droplet growth, as well as visco-elastic effects, which allow to relax elastic stresses.^{6} The latter effect can be captured by the theory of viscoelastic phase separation, which affects the coarsening behavior.^{30,31} This stress-relaxation, as well as the reduced stress due to hysteresis effects,^{15} could slow down elastic ripening. Finally, the droplets themselves can possess elastic properties. This is particularly true in biological condensates,^{32} which sometimes even form solid-like aggregates^{33} that potentially cause diseases.^{4} All these effects might be crucial for understanding the behavior of biomolecular condensates in cells.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/d0sm00182a |

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