Analytical model

An analytical model was developed to solidify the presented experimental findings. This model is based on Smoluchowski’s concept of fast coagulation [38], but can also be derived from Langmuir adsorption theory [124][125]. Smoluchowski first introduced the fast coagu-lation rateWK = 8πDRc as the rate at which two spherical particles of diffusion coefficient

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4. Static Structure Formation by DNA coated colloids

D, radius R and concentration c meet. Using the Stokes-Einstein equation D = _6πηR^kBT ,Wk

can be written as WK = ^4ck_3η^BT, losing its dependency on R. However, it is important to note that this is only valid in the limit of spherical particles. If anisotropic particles such as prolate ellipsoids are to be used in experiments, the rather complex dependency of D on the semi-principal axes of the ellipsoids [126] has to be taken into account, preventing further condensation of WK. Furthermore it has been shown in literature that also the hydrodynamic coupling of the spheres plays a significant role in aggregation rates. It can be estimated that this reduces the fast coagulation rate by a factor of µ = 0.54 [127].

For pseudo-irreversible processes such as they are investigated here, this is also the rate at which two spheres bind. In a simplified system, where one sphereA (minority sphere) with radius R is surrounded by an infinite number of majority spheres B with Radius R, that cannot bind themselves, but only the minority sphere A, the binding probability of A to B can be written as

WAB = 4cBkBTµ

3η Y(t) = WKY(t) , (4.1)

where N(t) is the number of B bound to A and Nmax is the maximum number of bound spheres bound to A. The accumulated number of B spheres onA with time is therefore

dN =WABdt (4.2) for the boundary conditions N(0) = 0. Reducing the number of B spheres from infinity to a finite reservoir of XB-A the concentration cB is effectively reduced in time by the factor θ= ^XB-A_X^−n(t)

B-A . Inserting this into equation 4.2 and subsequent solving for N(0) = 0 yields N(t) =N_max

In a regime where no multimerization occurs and the described accumulation process is the dominating process, equation 4.4 is a valid approximation for an ensemble of spheres. In this case,N(t) is the average number of accumulated B spheres onA spheres and N_max is the maximum of accumulated spheres onAspheres averaged over allAspheres. Consequently, XB-A is the stoichiometry betweenB and AparticlesXB-A = ^c_c^B

A. According to experiments, Nmax is best approximated by Nmax = 6.8 (see figure 4.6A). As the temperature and the viscosity of the sample can be measured experimentally, there are no remaining free variables in equation 4.4. Time course measurements atXB-A >90 for different concentrations show

4. Static Structure Formation by DNA coated colloids

Figure 4.6.: Analytic model for binary heterocoagulation at high stoichiometries. (A) Ex-perimental data shows that at a stoichiometry of XA−B = 110 binary heterocoagulation is reduced to aggregation of majority spheres on minority spheres. (B) Model predictions of equation 4.4 and experimental data for different microsphere volume fractions Φ. (C) Prediction of the model concerning the formation of isolated aggregation seeds (see equa-tion 4.8). The red line marks 50% free-binding sites on a minority particle on first contact with another minority particle. Consequently the stoichiometry Xgrowth = 22.3 can be con-sidered a critical stoichiometry for cluster growth. (Inset) Closeup of figure 4.2 with the estimated Xgrowth,exp in comparison to the predicted Xgrowth. Error bars denote the SD of the underlying cluster distribution.

that the model predictions of equation 4.4 are in good agreement with experimental data (see figure 4.6B) ,where only compact clusters with one minority sphere per cluster have been evaluated. Interestingly, also Langmuir adsorption theory leads to the same result. It has been shown elsewhere [125], that an irreversible adsorption of spheres from a finite reservoir to a surface can be described by

n(t) = nmax

1−exp(−kon(c0−cmax)t) 1− ^cmax_c

0 exp(−kon(c0−cmax)t)

!

. (4.5)

The Langmuir description lacks the explicit form of kon, but comparison to equation 4.4 yields

kon = WK

NmaxcB

= 4kBTµ 3ηNmax

. (4.6)

Hence it follows, that the binary heterocoagulation process can also be modeled by an irreversible Langmuir adsorption process with a purely kinetic on-rate, where the sum of

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4. Static Structure Formation by DNA coated colloids

surfaces of the minority spheres represents the total adsorbing surface.

As equation 4.4 describes the the accumulation of majority on minority spheres, the stoi-chiometry at which minority spheres are predominantly occupied by majority spheres when they meet other minority spheres on average for the first time can be estimated. This can be done by evaluating equation 4.4 at the fast aggregation timeτAA of minority spheres

τAA = 1

represents an approximation for the average free binding sites F on a minority sphere on first contact with another minority sphere (see figure 4.6C). The stoichiometry Xgrowth at which only 0.5 binding sites are available on first contact, can be calculated numerically, resulting in Xgrowth = 22.3. This value represents the critical stoichiometry at which more than half of the minority spheres are saturated on first contact, inhibiting further growth of the aggregates. To compare Xgrowth to experimental data, the following estimation can be used. As Xgrowth marks the point at which approximately 50% of aggregation seeds are able to merge to bigger clusters, it should in turn experimentally reflect the stoichiometry at which the mass average of the clusters is increased by 50% in relation to the compact regime of aggregation seeds. The data point X = 110 (see figure 4.2) and its error was chosen as a reference for the compact regime. Accordingly, a 50% increase of the mass average is found at Xgrowth,exp = 21.8±0.6, which is in good agreement with Xgrowth (see figure 4.6C, inset). As equation 4.8 is not dependent neither on the size of the spheres, nor of their concentration,Xgrowth can be considered a universal, scale-free constant, applicable to all binary spherical particle systems that are purely diffusion limited.