Aggregation kinetics of irreversible patches coupled with reversible isotropic interaction leading to chains, bundles and globules

Introduction

Biological particles self organize into highly monodisperse structures due to the presence of specific interaction sites on their surface [2]; examples include virus, proteins, etc. [3], [4], [5], [6], [7]. There are some specific kinds of proteins which aggregate together in ordered bundles leading to diseases such as cataract, Alzheimer’s disease and Parkinson’s disease [8]. In fact it has been shown that almost all neurodegenerative diseases occur due to the abnormal protein aggregation. These kinds of proteins are generally termed as amyloid proteins [9]. This type of bundling is also observed in semi-flexible polymers grafted on a surface [10]. In this particular case an attraction between the polymer chains leads to the collapse of the homogeneous phase to a bundled state.

To mimic some of these biological structures, colloidal particles which are asymmetric, patterned or patchy [11] have received considerable attention in recent years as their isotropic [12], [13], [14] counterpart was not able to explain some of the experimental observations like bundling. Particles which are anisotropic in shape or interactions are synthesized as they have very promising applications in electronics [15], drug delivery [16], [17], in fabricating photonic and plasmonic materials [18], [19], [20], [21], [22]. Several patchy particles have been developed where colloidal particles undergo DNA-mediated interactions and are known as DNA-functionalised colloids [23], [24], [25]. The patchy models have been already used to study the equilibrium properties of polymerization by Wertheim theory [26] where they developed a thermodynamic perturbation theory for patchy particles having two patch sites. Sciortino et al. [27] did an extensive study of the two patch site model and have shown that simulation matches with the predictions of Wertheim theory. In their study they have looked at the equilibrium structures formed when only one bond per patch was allowed. Structures such as tubes, lamellae are predicted in experimental and computer simulations by varying size and shape of the patches [28], [29], [30], [31], [33]. Although extensive work has been carried out for the case of patchy particles and inverse patchy colloidal particles [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], very few work has been carried out for the case where in addition to patchy interaction an isotropic potential is also considered along with it [45], [46], [47], [48], [49], [50]. These studies were confined to finding the equilibrium properties like liquid–liquid coexistence curve or to study the competition between phase separation and polymerization. Liu et al. [47] have observed that as the patch number decreases the critical point of the liquid–liquid binodal curve shifts towards the lower volume fraction and width of the binodal curve increases. Using these results they were able to explain the phase separation of proteins like lysozyme and α-crystalline molecule. Dudowicz and coworkers [45], [46], [48] studied the competition between phase separation and polymerization using lattice-based linear polymerization models. Audus et al. [50] recently studied the structural property of the equilibrium structure in a similar system, where they showed that the fractal dimension of this system turns out to be 2. It has already been shown that the reversible system is close to reaction limited aggregation model in the case of only isotropic potential and a fractal dimension of 2 [51]. Kastelic et al. [52] also studied a similar system where they estimated the number of patches and interaction strength of the patch using experimental data to reproduce the liquid–liquid coexistence curve, but in their study isotropic potential was not considered. In the present work, we have modeled a unique system where the bonds between the patches are irreversible while the bonds formed via isotropic potential are reversible. When two patches come in close contact they form a bond or in other words it follows diffusion limited cluster aggregation model [51], which has a fractal dimension of 1.8. As Prabhu et al. [53] have shown for monovalent patches, particles organize themselves into chains (fibers) and these chains form an arrested structure which consists of strands. Formation of fibers and bundles has also been observed by Huisman et al. [1] where they used an anisotropic Lennard–Jones potential to model two-patch particles. In this system above a particular temperature these particles form bundles, and they have shown that this transition is very similar to the sublimation transition for polymers. Also a transition from small clusters to long straight rigid tubes has been observed by Vissers et al. [31] in case of one-patch colloid of 30% surface coverage below a specific temperature which is density dependent.

In the present work we are using the simulation technique called the Brownian cluster dynamics (BCD). This method was developed for studying the structure, kinetics and diffusion of aggregating system of particles interacting via isotropic potential [32], [54], [57]. It has already been shown that BCD agrees with other Brownian dynamics simulations [56] and predicted phase diagrams are similar to Monte Carlo simulations [54], [57], [58]. The advantage over the other methods is that BCD can handle very large number of particles upto 10⁶ particles. Prabhu et al. [53] modified this method to accommodate asymmetric potential with isotropic potential on the particles. They have shown that when a single polymer chain was simulated using this technique they were able to get the correct static and dynamic properties ignoring hydrodynamic interactions. They went further and modeled the step growth polymerization where one patch could form only one irreversible bond under different solvent conditions where they observed that the kinetically driven system formed an arrested structure. In the present work we do not have any constraint over the number of bonds a patch can form, which leads to quite fascinating structures.

The paper is arranged in the following ways. In model and simulation techniques we briefly describe our simulation technique and also explain how we are changing the quality of the solvent. Then we present our results about the change in the structure of the system as we vary the patch size. For very small patches we form linear chains while on increasing the patch size we end up in a more globular structure, also the effect of changing the volume fraction of the system will be discussed. We will also discuss about the possible link between protein and patchy particle aggregation in the context of bundling, which is believed to be the origin of many neurodegenerative diseases, followed by conclusion in last section.

Model and simulation techniques

A model potential was developed by Kern and Frenkel [59] to simulate colloidal systems with anisotropic interactions. This model consists of hard spheres of diameter σ which is kept as unity with two patches, where the orientation of the patches is specified by a unit vector  v^i, which lies at the center of patch. The patch can be viewed as intersection of sphere with a cone of solid angle 2ω having vertex at the center of the sphere. We simulate hard spheres complemented with patch vectors v_i interacting through a square well patchy potential (U_iso(r_i,_j)) followed by Kern–Frenkel potential (Upatchy(r^i,j,  v^i,  v^j)). Here  r^i,j=ri,j/ri,j, where r_i,j is the vector connecting the center of mass of spheres i and j,v^i=vivi, v^j=vjvj are unit patch vectors of spheres i and j. The potential is given by

where ϵ is the interaction width which we have kept ϵ=0.1 and u₀ is the depth of the square well and σ is the diameter which is kept as unity in the present study. U_patchy depends on the orientation of the two particles and is defined as:

In the present study each patch vector is associated with two oppositely located patches and ω is a tunable parameter where we can change the percentage of the patchy surface coverage on the sphere. ω=90° corresponds to the irreversible aggregation of pure isotropic square-well potential and ω=0° corresponds to the hard sphere.

We start our simulation with an ensemble of N_tot randomly distributed spheres (each associated with a randomly oriented patch vector) of diameter σ=1 in a box of length L, where volume fraction ϕ=π6Ntot/L3. We have used a cubic box of fixed length L=50 with periodic boundary conditions in the present study. The simulation procedure involves two steps one is the cluster construction step and the other one is the movement step. In the cluster construction step, when two monomers or spheres are within the interaction range then a bond is formed with a probability α₀, while if a bond already exists then it is broken with a probability β₀, so that P=α0(α0+β0). Then P=1−exp(u₀/k_BT) is the relation connecting temperature T with the probability P as already shown by Prabhu et al. [53]. The collection of all bonded spheres together is defined as a cluster. For the square well system we have defined the reduced second virial coefficient B₂=B_rep−B_att, where B_rep=4, which is the repulsive part coming from hard core repulsive interaction between the spheres. B_att is the attractive part of the second virial coefficient due to the square well potential and is given by Batt=4. ⌊exp(−u0kBT)⌋.[(1+ϵ)3−1] [55], where T is the temperature and k_B is the Boltzmann factor, which is kept as unity in the present work. When two patches overlap the probability to form a bond is α_P=1 and the probability to break a bond is kept as β_P=0, thus forming irreversible bonds. In the present study we have allowed multiple bonds per patch which is quite different from the previous studies where they have employed an artificial constraint of single bond per patch. In the movement step we randomly select 2N_tot times a sphere then we either rotate it whereby we rotate the patch vector randomly with an angular displacement of s_R with respect to the patch vector or we translate the sphere with a small step size s_T in a random direction. Thus on average every particle undergoes both rotational and translational diffusion independently and in an uncorrelated manner as already demonstrated by Prabhu et al. [53]. If the rotation or the translation step leads to the breaking of bond or overlap with the other spheres that movement is rejected in our simulations. The step sizes have to be very small otherwise it will lead to a nonphysical movement due to rejection of the movement steps. It has already been shown for the parameters chosen in the present study, in order to obtain the correct diffusional behavior, the translation step size s_T=0.013 and the rotational step size s_R=0.018 are the best choices of parameters [53]. After a cluster construction and movement step is over the simulation time t_sim is incremented. The relation between physical time t and t_sim comes from the fact that for a free particle undergoing Brownian motion, mean squared displacement is given by 〈R2〉=6D1Tt, where D1T=1/6 is the translational diffusion coefficient of a single sphere. For a particle undergoing random walk 〈R2〉= tsimsT2 where s_T is a constant step size and t_sim is the number of simulation steps taken. The time taken for a sphere to travel its own diameter is given by t₀=⟨σ²⟩ therefore the reduced time is given by tt0= tsimsT2.

In the present work we have used three different B_att values:

• B_att=0: In this case system is in good solvent condition, there is no reversible isotropic interaction and the particles aggregate only due to irreversible bonding between the patches.

• B_att=4: As B_att increases, quality of solvent deteriorates or in other words reversible isotropic aggregation also contributes towards the structure. At this value of B_att the hard core repulsion is balanced by the attractive part of the potential and B₂=0. This condition corresponds to Boyle temperature of the fluid.

• B_att=12: For pure isotropic fluid, this B_att corresponds to the value where we observe liquid-crystal binodal [54] or can be considered as bad solvent condition.

We will be working with the above three B_att conditions, mainly for two different volume fractions ϕ =0.02 and ϕ=0.2 and for two different ω values 22.5° and 45°. As a result of irreversible patchy and reversible isotropic interaction we are able to define three different types of clusters (Fig. 1),

Fig. 1:

The red surface of sphere is hard sphere which undergoes only isotropic interaction and blue surface are the patches which undergoes anisotropic interaction. (a) Monomers forming irreversible bonds through patches (P type clusters). (b) Monomers interacting through reversible isotropic interaction (NPI type clusters).

• We define spheres which interact only through the patches, or spheres which are irreversibly connected to its neighbors as P cluster as shown in Fig. 1a.

• The spheres which only interact through the isotropic potential forms a different type of cluster which we call as NPI cluster see Fig. 1b.

• The clusters formed as a result of both patchy as well as isotropic interaction are called as PI cluster.

In the present work we run our simulation till the system forms a percolating cluster in PI construction, i.e. cluster that extends from one end of the box to the opposite end. We have done all the structural analysis in the present study after the formation of percolating cluster in PI construction.

Results and discussion

In the present work we have used two different patch angles ω=22.5° and ω=45° to study the aggregation of patchy particles mainly at two different volume fractions ϕ=0.02 and ϕ=0.2. The effect of the solvent condition was taken into account by changing the B_att as mentioned in previous section. In Fig. 2 we are showing the structures obtained from the simulations at B_att=0, B_att=4 and B_att=12 at two different ω values 22.5° and 45° for ϕ=0.02 after the same time t/t₀=1800, where the kinetics of the system no longer evolves. In Fig. 2aB_att=0 and ω=22.5°, we observe the formation of chains. As the patch angle is very small it is not possible to form more than one bond per patch due to hard core repulsion between the spheres, reversible isotropic potential also does not play any part in the aggregation process as B_att=0, so only chain formation is possible. In Fig. 2b we show the same system for B_att=4, a visual inspection itself reveals that the length of the chains is larger compared to B_att=0. This is due to the presence of isotropic attractive potential between the spheres, they stay in each other’s range for a longer period of time compared to B_att=0. As the particles diffuse within the bonds as the spheres also undergo rotational diffusion they are more likely to form irreversible bond through patches. For the case of B_att=12 we observe that the reversible part of the potential plays a major role which results in the transformation from chains to bundles as can be observed in Fig. 2c. In Fig. 2d we have shown the structure for B_att=0 for ω=45° where we observe the presence of dense clusters, as spheres are able to form multiple bonds per patch. For the case of B_att=4, the number of bonds per particle will increase as the reversible aggregation will also contribute towards the structure formation and we observe denser clusters. In the case of B_att=12 the clusters should have been denser, which we do not observe in the Fig. 2f as the dominant contribution for aggregation comes from the irreversible part of the potential. Irreversible aggregation always leads to branching and forms fractal type of aggregate.

Fig. 2:

Snapshots of the system at same physical time t/t₀=1800. From left to right, B_att is increasing from B_att=0, 4, 12 and the top picture is at ω=22.5° and bottom is at ω=45°.

Kinetics of aggregation

In order to understand the kinetics of aggregation we have followed the average number of bonded neighbors Z as a function of time for three different cluster construction type as explained in previous section. In Fig. 3 we have plotted the average number of bonded neighbor for the P construction (clusters formed only due to the patches) at ϕ=0.02 for different B_att values. The aggregation process starts from a randomly distributed spheres system and thus for all the three B_att we start at the same value of Z_P as the bonds are irreversible for the patches. If the distance between the center of mass of the sphere is within a distance of 1+ϵ and also the patches are aligned irreversible bonds are formed between the patches, while if patches are not aligned but are within the interaction range a reversible bond is formed. The average life time of such a reversible bond is given by 1/β, so as B_att increases the life time of the bond increases which means the sphere will be in each other’s interaction range for a longer period of time. The spheres stay in each other’s interaction range for a longer time and thus the patch vectors will align themselves thus forming an irreversible bond. As a result in the P construction the kinetics of aggregation becomes faster as we increase the attractive strength. We observe that for B_att=0, the number of bonded neighbors for P construction stagnates at a value ~2, which means for ω=22.5° on average only two bonds per particle are formed or in other words a patch is able to form only one bond. In the inset of Fig. 3 we have plotted the average number of neighbors which are connected to a sphere through both irreversible as well as reversible part of the potential (PI construction). For the case of B_att=0 and B_att=4 the average number of neighbors reaches a steady state value close to two signifying there is no densification in these systems. As the Z_PI~2 also signifies that for the PI construction as well, we have on average two bonds per particle and our system will have only chain like configuration for smaller values of B_att. For the case of B_att=12 we observe that the average number of neighbors increases to a value more than 2, indicating that we have a locally more dense system. This is expected because pure isotropic square well (system where we do not have anisotropic patchy interaction) counterpart undergoes phase separation through nucleation and growth phenomena at B_att=12 [54]. For comparison we have also plotted the case when we have pure square well potential undergoing irreversible DLCA [60] shown as dotted lines in the inset of the Fig. 3. Here we observe that the kinetics of aggregation for the case of irreversible pure isotropic square well and patchy particles are very different from each other indicating that isotropic reversible part of the potential is playing a major role in the aggregation phenomena as well as the structure.

Fig. 3:

The average bonded neighbors for P type cluster ⟨Z_P⟩ is plotted with respect to reduced time at different B_att values as indicated in the figure for ϕ=0.02 at ω=22.5°. The inset shows average bonded neighbors for PI type cluster ⟨Z_PI⟩ where the solid line is for irreversible DLCA system for the case of pure isotropic square well potential.

To understand the role played by the reversible part of the potential we have plotted in Fig. 4 the evolution of the average number of neighbors (Z_NPI) formed due to only the reversible part of the potential as a function of time for ϕ=0.02 and ω=22.5°. For B_att=0, we have α₀=0, thereby no reversible bonds are formed, while for B_att=4 and 12 we have both α₀>0 and β₀>0. For the case of B_att=4, the contribution of Z_NPI towards Z_PI is more compared to Z_P in initial time of the aggregation process t/t₀<0.2, which means the initial aggregation process is dominated by the reversible part of the potential and at later times attains a constant value 0.14. In Fig. 4 we also observe that the kinetics of aggregation of the reversible part of the patchy particle and pure isotropic square well potential are very similar, although for the patchy case the curve is always below the pure square well case as in this calculation we do not consider the particles that are bonded by irreversible bonds. After a time t/t₀>10, Z_NPI starts to decrease slightly as more and more particles are forming part of the P cluster as evident from Fig. 3, where Z_P keeps on increasing which means aggregation process is dominated by the irreversible aggregation of the patchy sites. For the case of B_att=12 we observe that Z_NPI follows similar trend as that of B_att=4 although the average number of neighbors are higher than B_att=4 as the attraction strength in the case of former is higher than latter. We also observe that after a time of t/t₀>30, the kinetics of aggregation is accelerated for the case of patchy particle at B_att=12, indicating a chain to bundle transition. For the case of pure isotropic potential this upturn indicates the gas-crystal phase separation which we do not observe in the case of Z_NPI as the system is stuck in the meta-stable state as shown by Babu et al. [54]. For the case of patchy particle we do not observe any kind of meta-stable state for chain to bundle transition.

Fig. 4:

The evolution of ⟨Z_NPI⟩ is plotted with respect to reduced time at ω=22.5° for ϕ=0.02 for B_att=4 (square) and B_att=12 (triangles). The filled symbols indicate irreversible patchy particles with reversible isotropic interaction and open symbols indicate reversible aggregation of purely isotropic square well potential.

As observed in the irreversible aggregation of pure square well fluids, the volume fraction of the system also plays a major role in deciding the kinetics and structure of the patchy particle aggregate [60]. In Fig. 5a we have plotted the evolution of the average number of neighbors due to the patchy part of the potential Z_P as a function of time for ϕ=0.02 and ϕ=0.2 at ω=22.5°. As we increase the volume fraction of the system the number of particles with their bond vector oriented against each other as well as the number of particles within each other’s range increases. This is the reason why Z_P starts at a higher value for ϕ=0.2 compared to ϕ=0.02 where we have started both the aggregation process from a random system. For the case of P construction we observe Z_P~2 indicating the fact that for higher ϕ also on average the patches are able to form only one bond per patch when ω=22.5°. The value of Z_P also reaches a steady state value of ~2 for B_att=4 and B_att⁼12 but is slightly above for B_att=0 indicating the fact that more number of patches have formed bonds thereby the clusters have increased in size. For the case of NPI construction see Fig. 5b the number of neighbors Z_NPI for B_att=4 rises initially and then goes down before attaining a steady state value quite similar to the case of ϕ=0.02. When the spheres are in each other’s interaction range, they may form a reversible bond and then they diffuse within the bonds such that their patches get aligned whereby they will form irreversible patchy bond, thus the number of reversible bonds reduces. In the case of higher volume fraction and weaker reversible attraction B_att=4, we form chains and the chain length is higher as compared to the case of B_att=12 which forms bundles. For the case of B_att=12 even though the upturn in Z_NPI is not as evident as in the case of ϕ=0.02, still this system tries to densify but it is hindered as the movement of the spheres are restricted due to the increased number of particles in the system.

Fig. 5:

Comparison between the average bonded neighbors for ϕ=0.02 (closed symbol) and ϕ=0.2 (open symbol) at B_att=0 (circle), B_att=4 (square) and B_att=12 (triangle) for ω=22.5°. (a) The average number of bonded neighbor for P type cluster ⟨Z_P⟩ is plotted as function of reduced time t/t₀. (b) The average number of bonded neighbor for NPI type cluster ⟨Z_NPI⟩ is plotted as function of reduced time t/t₀. The double dotted line indicates the value when the average number of bonded neighbor is 2.

As explained in previous section in our simulation we are allowing multiple bonds per patch, which means the patch angle will also play a significant role in the kinetics as well as the structure that is formed during the aggregation process. In Fig. 6 we have plotted the Z_P as a function of the reduced time for the case ω=45° for ϕ=0.02. B_att=0 has the slower kinetics and attains a steady state value for t/t₀>50. As expected B_att=12 has higher number of patchy neighbors compared to the other B_att values, as the spheres are connected through the reversible part of the potential and hence they are more likely to diffuse within bonds to form irreversible patchy bonds. In all the three cases Z_P attains a steady state value greater than 2 indicating multiple bonds are formed per patch. As Z_P>2 we are not observing any chain formation for higher ω value but more globular type structure. In the inset we have shown the evolution of Z_PI as a function of time, in all the cases it goes to the same steady state value which means that for higher patch angles the irreversible part of the potential dominates compared to ω=22.5°. For earlier time Z_PI for B_att=4 and B_att=12 starts from a higher value compared to B_att=0 as the number of bonded neighbors increases due to the reversible interaction of the potential. For the case of B_att>0 the value of Z_PI will always be smaller than for the case of pure isotropic irreversible DLCA. Although the evolution follows very closely the pure isotropic DLCA, indicating that system wants to evolve the same way, but is restricted due to the finite patch size. On increasing ω further we will converge towards the irreversible pure isotropic DLCA system [60].

Fig. 6:

⟨Z_P⟩ is plotted as a function of reduced time for ϕ=0.02 at ω=45° when B_att=0 (circle), B_att=4 (square) and B_att=12 (triangle). The inset shows the average number of neighbors for PI type cluster and the dotted line indicates the irreversible DLCA aggregation when only isotropic interaction is present.

In Fig. 7 we have plotted Z_NPI as a function of reduced time for ϕ=0.02 for the case of a pure square well reversibly aggregating system and only the reversible part of the patchy particle with ω=45°. We observe that in the initial times the contribution of the isotropic part of the potential is not significant compared to the irreversible aggregation of the patchy particles as the maximum steady state value is less than 0.9, which means on average less than one particle is reversibly connected to each particle. The particles which were connected through the reversible part of the potential later diffuses and forms irreversible patchy bonds, which is the reason for the dip we are observing in Z_NPI. These small clusters formed will diffuse and further densify into a globular form thereby Z_NPI starts to increase even though the rate of increase is very slow. The increase in Z_NPI for ω=45° is characterized by the increase in the size of the clusters as observed in Fig. 2e, whereas for ω=22.5° the sudden increase in Z_NPI leads to chain to bundle transition.

Fig. 7:

⟨Z_NPI⟩ is plotted as a function of time t/t₀ for ϕ=0.02 when the value of ω=45° at B_att=4 (square) and B_att=12 (triangle). The filled symbols indicate the system with both irreversible patchy and reversible isotropic interaction and open symbols indicate reversible purely isotropic square well potential.

We have already shown that for ω=22.5° chains are formed irrespective of the B_att used in the system. In Fig. 8 we have plotted the chain length as a function of time for a volume fraction ϕ=0.02. For calculating the chain length we identify the sphere with only one bonded neighbor through the patches, the neighboring sphere will have two neighbors through the patches if it is connected to a chain. We keep on counting the connected neighbor through the patches till we reach a monomer with only one bond, which will be the opposite end of the chain. The average chain length is defined as ⟨l⟩=∑m_lN(m_l)/∑N(m_l), where m_l is the mass of the chain and N(m_l) is the size distribution of chain lengths. For the case of B_att=0 the average chain length reaches a constant value 9.7, which is consistent with the predictions of Sciortino et al. [27]. While for the case of B_att>0 studied in the system the chain length goes on increasing, indicating that as the reversible interaction of the particles increases the length of the chain is larger at the initial times, as can be observed when we compare the Figs. 3 and 4. At longer times we observe that the chain length for B_att=4 crosses over B_att=12, as well as B_att=20 and B_att=12 crosses over B_att=20 i.e. on average we have longer chains for the case of smaller isotropic attraction. This may seem contour intuitive, but what we observe for B_att=12 and 20, there is a chain to bundle transition, while for B_att=4 the system remains as chains. For B_att=4 the particles or chains get attached to the other chains and then they have enough time either to break away or diffuse to form irreversible patchy bonds thereby increasing the chain length. For B_att=12 and 20, we observe similar kinetics but as the attraction is higher between the particles, once they form part of the chain they start to aggregate as bundles.

Fig. 8:

The growth of average chain length as a function of time at ω=22.5° and ϕ=0.02 for different B_att as indicated in the figure.

Structural analysis

To identify where the spheres are distributed in the patches we have calculated the probability of occurrence of an angle for a single particle g(θ) between the patch vectors of two particles (the area under the curve has been normalized to one). If both the patches face in the same direction then the angle between the patch vectors is zero while if they face each other the angle between them is defined as 180°. The distribution of the angle defined between all the patch vectors in the system g(θ) is quite similar to the pair correlation function where we use positions instead of angles. In Fig. 9 we observe that for B_att=0 and B_att=4 for ω=22.5° at ϕ=0.02 only two angles are most probable 20° and 160°. We already know that Z_PI<2 which means that on average we form one bond per patch or in other words we have chain conformation. We also observe that around the angle of 40°, g(θ) converges to zero faster for B_att=0 and B_att=4 compared to B_att=12. A tale is developed for the distribution of B_att=12 close to 40° and 140° which is the signature of bundling at very low concentration. Tavares et al. [61] have shown that for the case of B_att=0 the most probable angle that they observe for the case of 22.5° turns out to be 22° which agrees with our results as well.

Fig. 9:

g(θ) the probability distribution of the angle for a single sphere due to the bonded particle is plotted at different B_att as indicated at ω=22.5°. The arrows indicate the extended tail of the distribution indicating a chain to bundle transition.

In Fig. 10 we have plotted the distribution of g(θ) for three different ϕ and B_att values at ω=22.5°. Here we observe that even for B_att=0 all the angles between 40° and 140° are possible for moderate to high volume fractions. As the volume fraction increases the rearrangement of the patches as well as the diffusion of the cluster for aggregation becomes difficult, as it may lead to bond breakage or may lead to overlap with the neighboring spheres. Also the maximum for the most probable angle that we observed for low volume fraction comes down and shifts inwards for moderate and high ϕ values as the particles maximize the reversible bonds thus going to a lower energy state. In this case we also observe another peak which appears at 90° which means the particles form bundles and these bundles branch out similar to the spaghetti like structure.

Fig. 10:

The probability distribution g(θ) is plotted for B_att=0 (open symbols) and B_att=12 (closed symbols) at ω=22.5° for ϕ=0.02 (triangle), ϕ=0.2 (square) and ϕ=0.4 (circle).

In the present study even though multiple bonds are allowed for the patches, ω=22.5° was able to form only one bond per patch. For the case of ω=45° as shown in Fig. 11 we observe that peaks which existed for the case of ω=22.5° are not very prominent and that the maximum has shifted to 60° and 120° for all the B_att and ϕ values. This can be understood from the fact that for ω=45° we are allowing the patches to form multiple bonds. The patchy particles will try to maximize the irreversible bonds so the most probable angle changes from the case of ω=22.5°. It has already been shown that the average number of bonded particles per patch increases from 2 to 4 see Fig. 6. As the attraction increases there is another probable angle coming up at 90° for all the volume fractions we have studied. This is due to the reversible part of the potential, as for B_att=0 (also see Fig. 10) for all the volume fraction we have considered in the present study we observe a minimum at 90°. Thereby we can conclude that reversible part of the potential favors branching of the clusters as observed for the case of ω=22.5°.

Fig. 11:

The probability distribution g(θ) is plotted for ϕ=0.02 at ω=45° for a range of B_att as indicated.

In the present work we have shown that for the case of ω=22.5°, the patchy particles always form chains for B_att=0 and B_att=4. Whereas at B_att=12 we observe that chains aggregate together to form bundles. It was suggested by Huisman et al. [1] that this transformation is similar to the sublimation transition of polymers. They have shown that the transition happens at a specific temperature using Monte-Carlo simulations. In the present work we also observe a very sharp transition from chains to bundles for the case of ω=22.5° between B_att=4 and 6. The kinetics of aggregation in the case of NPI cluster construction gives us a clear indication that the transition is similar to a nucleation and growth type mechanism. We also observe that the chain length at B_att=4 where only chains are formed, is higher than for the case of B_att=12 where bundling happens. This is contrary to the work of Huisman et al. [1], because in their case the individual particle can break and form bonds, while in the present study we have irreversible bonds for the patches. So once the bonds are formed among the patch they grow as chains for the case of ω=22.5° and these chains aggregate together to form bundles. In order for the chain to grow they have to align themselves and the probability for two chains to align is very small, due to the hindrance created by other cluster around.

The chain to bundle transition is also believed to be the reason for many neurodegenerative diseases like Alzheimer’s. There the monomers aggregate to form oligomers which then transfer to bundle configuration commonly called amyloid fibers. These fibers are chemically stable quite similar to our present model where irreversible bonds give structures formed a stable conformation. These fibers then aggregate together to form a percolating cluster or gels depending on the pH of the solution, which is very similar to the present model where we started with monomer which aggregate together to form chains which transformed into a bundle depending on the interaction strength or the quality of the solvent. These amyloid fibers form helical bundles similar to the present study for individual arm of the gel formed for the case of ϕ=0.02, ω=22.5° at B_att=12 see Fig. 3c. The present model gives a good qualitative description of the different process involved in the amyloid fiber formation more quantitative study will be reported in the future.

Conclusions

We have used Brownian cluster dynamics to investigate the kinetics of aggregation and structure of the resulting aggregates for the two patch system by varying patch size and solvent condition. In the model studied, the anisotropic potential is complemented by square well isotropic potential, where we have also defined three different definition of clusters. We have observed that for small patches we form chain like configuration and also the average chain length increases on increasing the isotropic interaction from B_att=0 to B_att=4. For B_att=12 and 20 the average chain length is less than B_att=4 as a result of the chain to bundle transition via nucleation and growth mechanism. When the size of the patch was increased to ω=45° we observed globular structure instead of the chains like configuration. We have shown that in the case of small patch size (ω=22.5°) the structure and kinetics of aggregation is dominated by the isotropic reversible interaction while for larger patch size (ω=45°) it is dominated by irreversible patchy interaction and on further increasing the patch size the system will tend towards the isotropic irreversible DLCA aggregation. In the case of ω=22.5° the bond distribution around a single particle showed that only two bond angles were most probable indicating chain formation. On further increasing the attraction the distribution developed a tail indicating chain to bundle transition. While for the case of ω=45° we observed that there is a contribution from other angles as well giving us a more globular conformation. It will be interesting to study how the kinetics of aggregation and structure of aggregates changes on making the anisotropic interactions also reversible which will be addressed in future work.