< Introduction >

A gel is a space-spanning network and its dynamics is macroscopically arrested. Thus a gel has static elasticity and is widely used in our daily life. A gel has attracted considerable attention from the fundamental viewpoint because of its intriguing non-equilibrium and non-ergodic nature.

Among model gel systems, colloidal gels have been widely used because they provide a single particle resolution and tunability of inter-colloid attraction. Colloidal gels are formed by short-range attraction between colloids, and colloid-polymer mixtures have been often used. In a colloid-polymer mixture, colloids attract each other to increase the configurational entropy of polymers. We can control the attraction strength by the polymer concentration and its interaction range by the molecular weight of the polymers. Previous works have focused on both the phase diagram and the formation process of colloidal gels.

The phase diagram of colloidal gels is determined by two physical parameters: the colloid volume fraction and the polymer concentration which plays a role of the effective temperature (or the attraction strength). Experimentally a gel is formed as the attraction strength becomes strong enough to cause spinodal decomposition. Previous works have focused on the relation between gelation and spinodal decomposition. For example, Lu et al. measured the gelation boundary precisely and showed that it indeed coincides with the boundary of spinodal decomposition [Lu et al., Nature, 2008].

The mechanism of the gelation process is still controversial and simulation works suggest that hydrodynamic interactions play key roles in network formation. In Brownian dynamic simulations, systems form droplet structures in moderate strengths of attraction and gelation is induced only when we introduce attraction much stronger than the corresponding experimental value. In Brownian dynamic simulations only thermal fluctuations and short-range attraction are considered and the effects of hydrodynamic interactions are completely ignored. To study the effects of hydrodynamic interactions on gelation, Araki and Tanaka have developed the new simulation method for colloidal suspensions that can properly incorporate many-body hydrodynamic interactions and reported that hydrodynamic interactions strongly enhances network formation [Araki and Tanaka, PRL, 2000].

However, there has been little experimental evidence for the roles of hydrodynamic interactions in gelation processes because previous experiments could not access the early stage of gelation process. Here, we develop a new experimental method which allows us to directly follow the whole process of gelation with a single particle resolution and study the mechanism of gelation using it.

< Materials and methods >

Previous experimental works have focused on the 'final structures' of colloidal gels. In the previous method, all components are mixed in a sample bottle, incubated into a glass cell, and then observed by a microscope. This method has two fundamental problems for direct observation of the gelation process. First, gelation already starts in the sample bottle, which prevents the observation of the early stage of the gelation. Second, the process of inserting the sample to the glass cell causes strong convective flow and strongly perturbs colloid motion. Therefore, the previous method is not suitable for direct observation of the kinetics of gelation.

This thesis focuses on the gelation process and its mechanism. So, we developed a new method. Gelation requires three components: colloids, polymers for inducing short-range attraction between colloids, and salt for screening the surface charges of colloids. Without salt, colloids are well dispersed by long-range electrostatic repulsions. Addition of salt reduces electrostatic repulsions and initiates aggregation of the colloids under the action of polymer-induced short-range attractions. In our method, a sample is initially prepared without salt and incubated into an observation cell attached with a membrane filter. After observation is started, a salt-reservoir is contacted with the observation cell. Then salt diffuses into the observation cell through the membrane filter, which initiates gelation. Our method allows us to observe the entire process of the gelation from the very early stage to the late stage without suffering from convective flow.

We synthesized sterically stabilized poly-methyl-methacrylate colloids with their radii of about 1.4 um. Colloids were fluorescently labeled and observed three-dimensionally by a laser-scanning confocal microscope. We calculated the center positions of all colloids from the three-dimensional images with a help of a computer and analyze the structure and dynamics of the colloidal suspension.

< Cluster aggregation >

In results and discussions, we focus on four subjects: cluster aggregation, gelation in a dilute system, gelation in a dense system, and dynamical arrest. Cluster aggregation occurs in the early stage of all gel samples and each cluster forms a 'building block' in the gel networks. Therefore, understanding cluster aggregation is the basis for the gelation mechanism.

Previous simulation works have shown the effects of hydrodynamic interactions on cluster aggregation. Furukawa and Tanaka have prepared 13 colloids in an isosahedral configuration and let them aggregate. They have compared results of Brownian dynamics (without hydrodynamic interactions) with those of Fluid Particle Dynamics (with hydrodynamic interactions) [Furukawa and Tanaka, PRL, 2010]. In Brownian dynamics simulations, the colloids aggregate and form a compact cluster within a Brownian time of a colloid. In Fluid Particle Dynamics simulations, on the other hand, the fluid between the colloids must be squeezed out, which slows down cluster compaction by more than 10 times. This is a consequence of the incompressible nature of the solvent. Therefore, we focus on compaction dynamics of small clusters experimentally to elucidate the effects of hydrodynamic interactions.

We prepared a sample of the colloid volume fraction of 5 % and the polymer concentration of 1.0 mg/ml. As aggregation proceeds, the averaged cluster size and number of nearest neighbors (coordination number) monotonically increase. We successfully observed that three colloids are first connected linearly and then gradually form a compact cluster. All clusters with three colloids are identified and tracked with a high time resolution. From the measurements of the temporal change in the radius gyration of each cluster, we found that the process of cluster compaction takes more than 10 times of the Brownian time. The clusters composed of four and five colloids also show similar results. These results strongly suggest that hydrodynamic interactions severely retard cluster compaction and lead to long-lived linear open clusters.

To understand aggregation of the clusters that have more than five colloids, we evaluate the cluster structures by calculating the fractal dimension df. In the early stage of aggregation, df is around 1.4. As aggregation proceeds, df increases to 1.8. Therefore, the clusters form chain-like structures (df = 1) in the early stage and gradually form compact structures (df = 3). The clusters of df = 3 are energetically favored, whereas the clusters of df = 1 is kinetically formed due to hydrodynamic interactions. Thus, we may say that hydrodynamic interactions compete with local energy minimization in each cluster.

Our results suggest that hydrodynamic interactions dominate the early stage of cluster aggregation, namely, the kinetic pathway is selected by momentum conservation.

< Gelation in a dilute system >

To study gelation in a dilute system, the colloid volume fractions are fixed at 7 % and only the polymer concentrations are changed. In each gel sample, both cluster size and coordination number increase as gelation proceeds. The radius gyration of the largest cluster increases to the system length, which leads to percolation. The fractal dimension increases from 1.4 in the early stage of gelation to 1.8 in the late stage. These results are consistent with the above results of cluster aggregation.

The question here is how a dilute gel is able to form an open space-spanning network. In our system, open clusters are connected with each other and finally form a space-spanning network. Because df is lower than three and increases as gelation proceeds, the colloid volume fraction itself has little importance. Instead, we calculate the 'effective' volume fraction by replacing each cluster to the effective sphere whose radius is equal to the radius gyration of the cluster. In all gel samples, the effective volume fractions sharply increase to approximately 60 %, which leads to percolation.

Our results reasonably explain the mechanism of percolation in a dilute gel. Hydrodynamic interactions decrease the fractal dimension of clusters down to 1.4 - 1.8. Thus the effective volume fraction can increase up to 60 %, which leads to percolation. If there were no hydrodynamic interactions, the fractal dimension would become close to three, the effective volume fraction could not increase, and the system would form droplet structures instead of a space-spanning network. Our finding suggests that hydrodynamic interactions play a key role in the percolation in a dilute gel.

< Gelation in a dense system >

To study gelation in a dense system, the colloid volume fractions are fixed at 15 % and 30 % and the polymer concentrations are changed. In all samples, the percolation processes are accomplished in a few minutes and then the networks slowly coarsen over several hours. The percolation process hardly depends on the polymer concentration while the coarsening process strongly does.

When the system percolates, the coordination number is approximately three and the bond-bond angle is approximately 100 deg. Thus, the percolated network has chain-like structures. Our results of cluster aggregation suggest that hydrodynamic interactions dominate the system in the early stage and facilitate the formation of linear open clusters. Because the system has a high colloid volume fraction, the linear clusters connect each other before they transform into compact structures. Therefore, the clusters form a chain-like network. This scenario is consistent with the process predicted by Fluid Particle Dynamics simulations. Our results suggest that hydrodynamic interactions also play key roles in the percolation in a dense gel.

In contrast, the coarsening process depends on the polymer concentration. As the polymer concentration decreases, the coordination number increases more rapidly. These results suggest that local energy landscape plays key roles in the coarsening process. With lower polymer concentration (or weaker attraction), the energy barrier for motion of each colloid is shallower and colloids more easily move to minimize the free energy. As the coordination number increases, energy barriers become higher and finally trap colloids. Furthermore, we directly observe the elementary processes of network coarsening. Chain-like rings in a percolated network gradually shrink and form compact domains with a help of thermal fluctuations of each colloid. In some cases, tension accumulated in the network leads to disconnection of weak chains. These continuous and discrete processes of network coarsening provide direct information on how the system lowers the free energy in the coarsening process.

< Dynamical arrest >

Dynamics of colloids is arrested in a gel. Previous works studied the structures and dynamics of colloidal gels and suggested that local condensation causes dynamical arrest. However, the 'process' of dynamical arrest has been poorly understood. Here, we directly investigate the relation between local condensation and dynamical arrest in gel samples with the colloid volume fraction of 7, 15, and 30 %. Even when the systems accomplish percolation, results of the analysis of the mean square displacements show that colloids are still mobile. Over several hours after percolation, the coordination numbers slowly increase, which leads to the decrease in the mean square displacements. Therefore, our results directly show a causal link between local condensation and dynamical arrest in a gel.

< Conclusion >

Gelation processes have been poorly understood experimentally. Here, we develop the new method and directly observe the entire processes of the gelation with a single particle resolution. Our results suggest that coupling of hydrodynamic interactions and local energy minimization plays a crucial role in gelation and ageing. Hydrodynamic interactions disturb compaction of colloids, enhance the open nature of the cluster structure, and help percolation. On the other hand, local energy minimization is a driving force of local compaction, which plays key roles in the coarsening process and dynamical arrest.

Our study sheds new light on the aggregation mechanism of not only colloidal suspensions but also similar complex fluids made of dynamically asymmetric components, such as polymer solutions and protein solutions.

本研究は、ゲルの形成メカニズムを単一粒子レベルで明らかにすることを目的とし、コロイド・高分子混合系がゲル化する過程を共焦点レーザー顕微鏡によって実空間・実時間観察し、流体力学的相互作用がネットワーク形成に果たしている役割を実験的に示した。

第1章では、研究背景と目的について記されている。ゲル状態とは、3次元的なネットワークが形成され、かつ、その運動が凍結した空間的不均一状態である。ゲルは非平衡・非エルゴード状態であるため、その形成メカニズムについて大きな注目が集まっていた。数値計算を用いた先行研究では、流体力学的相互作用を取り入れるとネットワーク形成が大きく促進される効果が示されていた。しかしながら、従来の実験手法ではゲルの形成プロセス(特に初期過程)を観察することが原理的に困難であり、ゲルの形成過程について実験的な裏付けが乏しいことが問題であった。

第2章では、新しい実験手法の開発を中心として、蛍光コロイドの合成、共焦点レーザー顕微鏡の原理、単一粒子レベルでの解析手法について記されている。本研究を通して新たに開発された手法では、静電遮蔽によりゲル化を誘起する塩を、顕微鏡観察を始めた後に浸透拡散によって取り込ませる。ゲル化が塩の浸透により流れ場を誘起することなく準静的に開始されるため、ゲル形成の初期過程を含む全過程を単一粒子レベルで観察することに初めて成功した。

第3章では、コロイド濃度が希薄な試料(体積分率5%)について、クラスターの形成メカニズムについて記されている。クラスターが成長する過程において構造のコンパクトさを評価するために、フラクタル次元を調べた結果、1.4 ~ 1.8であり紐状の構造を持つことが明らかになった。コンパクトでない(エネルギー的に安定でない)構造を作るメカニズムとして、流体力学的相互作用が重要な役割を果たしていることが示唆された。流体力学的相互作用は溶媒の非圧縮性を起源としている。クラスターがコンパクト化する際には内部の溶媒を外部に絞り出す必要があり、その過程が律速となってクラスターの構造がコンパクトでない状態に長時間留まっていると考察された。

第4章では、コロイド濃度が比較的低い試料(体積分率7%)においてゲル化のメカニズムについて記されている。ゲル化の初期過程においては、クラスター形成の場合と同様、コンパクトでないクラスターが形成され、そのフラクタル次元は1.4 ~ 1.8程度であった。それらのクラスターが結合を繰り返すことでネットワークが形成されることが分かった。我々は、クラスターのフラクタル次元が3次元よりも大幅に低いことに注目して、系の"実効的な"体積分率を評価するために、全てのクラスターをその慣性半径と同じだけの大きさを持った剛体球に置き換えをした後で、実効的な体積分率を計算した。その結果、ゲル化の進行につれて、実効的な体積分率が初期の7%から60%に向かって大きく増加した。第3章の議論を併せると、流体力学的相互作用がクラスターのフラクタル次元を下げ、その結果クラスターの成長と共に系の実効的な体積分率が急激に増加し、実効的な体積分率が60%程度に到達することで、系はパーコレートしたネットワークを形成するという描像が得られた。これにより、これまで数値計算によって提唱されてきた流体力学的相互作用の役割を、はじめて実験的に検証することに成功した。

第5章では、コロイド濃度が高い試料(体積分率15~30%)におけるゲル化のメカニズムについて記されている。ゲル化の初期過程において、コロイドは直線的な構造を持ったクラスターを形成し、それらが繋がって短時間(数分間)の間に鎖状のネットワークが完成された。次に、長い時間(数時間)に渡ってネットワークの粗大化が起こり、ローカル構造がコンパクト化し、最終的に粒子の運動が凍結した。第3章で議論したクラスター形成のメカニズムに基づくと、流体力学的相互作用が初期クラスターのコンパクト化を阻害し直線的な構造形成を促進しているものと考えられる。さらに、流体力学的相互作用を考慮した数値計算が予測した構造パターンと実験結果は非常に良い一致を見せた。よって、濃いゲルにおいても、流体力学的相互作用がネットワーク形成を促進する役割を担っていると考えられる。一方、粗大化の過程においては、個々のコロイドが熱揺らぎを使って局所的にコンパクトな構造を形成することが分かった。より詳細に素過程を調べたところ、ネットワークの一部が張力によって切断される過程や、ネットワークの中にあるリング状の構造が収縮してコンパクトなクラスタードメインを形成する過程が直接観察された。

第6章では、ゲル内部の運動と構造について解析について述べられている。ネットワークを形成するだけでは運動の凍結には不十分であり、局所構造のコンパクト化が運動の凍結をもたらしていることが示された。

第7章には、論文の結論が記され、さらに一般論として、動的に非対称な2元混合系において流体力学的相互作用が重要な役割を果たし得ることが記されている。

以上のように、本研究で得られた成果は、ゲル形成の全過程を単一粒子レベルで観察した世界初の事例であり、流体力学的相互作用の役割を中心として、ゲル化のメカニズムの理解を大きく前進させた物理工学上重要なものである。よって本論文は博士(工学)の学位請求論文として合格と認められる。