M.
Danny Raj
and
Raghunathan
Rengaswamy
*

Indian Institute of Technology, Madras, 150, Mechanical Sciences Block, Chennai-600036, India. E-mail: raghur@iitm.ac.in; danny.presto@gmail.com

Received
31st July 2015
, Accepted 25th September 2015

First published on 25th September 2015

A single coalescence event in a 2D concentrated emulsion in a microchannel can trigger an avalanche of similar events that can destabilize the entire assembly of drops. The sensitive dependence of the process on numerous parameters makes the propagation dynamics appear probabilistic. In this article, a stochastic simulation framework is proposed to understand this collective behavior in a system employing a large number of drops. We discover that the coalescence propagation dynamics exhibit a critical behavior where two outcomes are favored: no avalanche and large avalanches. Our analysis reveals that this behavior is a result of the inherent autocatalytic nature of the process. The effect of the aspect ratio of the drop assembly on the propagation dynamics is studied. We generate a parametric plot that shows the region of the parameter space where the propagation, averaged over the ensemble, is autocatalytic: where the possibility of near destabilization of the drop assembly appears.

We adopt a complex systems approach – where drops can be modelled as individual entities interacting with each other and the surroundings^{20} – to solve the problem. Such an approach has been used for modelling other physical phenomena that involve non-coalescing droplet motion in droplet microfluidics.

Examples would be phonon-like behaviour because of long range interactions,^{21} self-organization into ordered patterns,^{22} and discrete droplet decisions in microfluidic loops that lead to interesting temporal patterns.^{23} In these cases, models that explain drop-level interactions are developed and the system-level behaviour is simulated by considering the effect of all the drops together.^{24–27} However, not all the interactions are easy to model from a first principles approach. For example, Bremond et al. in their experiments realized that the sensitive dependence of coalescence on local conditions made the process appear stochastic. We make one of the first attempts to study such problems in droplet microfluidics through a stochastic framework. In this article, we address the problem of coalescence propagation observed by Bremond and co-workers, where we try to understand the coalescence propagation in a 2D ensemble in a Hele-Shaw geometry. Drops can be packed tightly in such a microchannel to form a 2D concentrated emulsion. The stability of the emulsion depends on its immunity to coalescence events. If local coalescence events propagate through the assembly of drops, stability is compromised. Hence it is important to understand the collective behavior of drops in 2D microchannels.

Fig. 1 (a) Experimental evidence of avalanches (snapshots) of different sizes (see inside box) (Bremond et al., 2011) [courtesy: Nicholas Bremond for sharing video results not available in the literature]; (b) probability of coalescence as a function of theta (data digitized from ref. 19), fit-polynomial (ref. 19) and P(θ) according to eqn (1); (c) a realization in the stochastic simulation that shows very little propagation, (d) realization showing high levels of propagation. [Black: initial pair that is made to coalesce; green drops connected by yellow lines: drops that have coalesced; peach: free drops that are yet to coalesce.] |

(1) |

(ii) Coalescence is initiated through a randomly chosen pair of drops i and j, which are represented as circles with a black fill in Fig. 1c and d.

(iii) Drops i and j are assigned to the set Ac (active drops) in the following manner: Ac = {(i,j), (j,i)}, where the first element of the ordered pair represents the node from which the propagation is about to happen and the second element is used to determine the angle made by the pair with their neighbouring drops.

(iv) A neighbour set Nc is generated, which is represented as Nc = {(n^{1}_{i,j}, n^{2}_{i,j}…n^{p}_{i,j}), (n^{1}_{j,i}, n^{2}_{j,i}…n^{q}_{j,i})}; p, q < 5, a set of n-tuples where n can take a maximum of 5. Each list of n-tuples corresponds to a specific ordered pair in Ac. Definition of a neighbour: drop 1 is the neighbour of drop 2 if the distance between them in the lattice is equal to the sum of the radii of the drops 1 and 2.

(v) If Nc ≠ ∅ continue; else set Ac = ∅ and go to step (vii).

(vi) Coalescence propagation through each of the elements (drops) of the set Nc is effected stochastically using the probability P(θ),

(a) choose a drop (k) from the set Nc

(b) compute the probability based on its orientation with the corresponding ordered pair in set Ac

(c) select a uniform random number U ∈ [0, 1]

(d) if U < P(θ_{k}), then k has coalesced with the cluster – add this drop to a new set Ac_new

(e) repeat the steps for all drops in all the n-tuples in the set Nc.

(vii) Set Ac = ∅ and assign Ac = Ac_new; if Ac ≠ ∅ go to step (iii), else continue.

(viii) Terminate.

A randomly chosen coalescing pair of drops has ten neighbours (if the pair is not at an edge of the lattice). Propagation can happen through all these neighbours. But the stochastic nature of the problem reduces the chances of propagation through all neighbours based on P(θ). The framework that we propose, as explained in the algorithm, effectively uses P(θ) that explains coalescence locally to simulate propagation in the entire assembly.

In an attempt to understand the appearance of the hump in the P_{A}(Av), we try to investigate why certain avalanches in the size range ∼(0.1–0.3) have a lower probability of occurrence. To do so, we observe the average ensemble behaviour at the last generation and the dynamics near the avalanche size that has the minimum probability. As coalescence propagates through a sequence of events, the size of Ac in the cluster varies. In the last generation (event) there are M_{L} drops in the set Ac, which have to stop coalescing for propagation to cease. From Fig. 3a, it can be seen that it is highly likely that a small M_{L} will be found. From Fig. 3b it is also evident that the probability of finding enough neighbours in the last generation follows a similar trend. So for a typical avalanche, the active drops have to reduce to a small number before the propagation stops. Av_{min} is the point at which P_{A}(Av) shows a minimum in Fig. 2c. From the dynamics of the avalanche process, we compute the average number of events (number of generations) taken for a growing avalanche to reach (or cross) the size (Av_{min}) with minimum probability. From Fig. 3c one can infer that avalanches that stopped growing when its size was close to the minima (Av_{min}) take longer time to reach Av_{min} than the larger avalanches that have higher probability of occurrence. This means that the rate of cluster growth of an avalanche when its size is Av_{min} is higher for avalanches with higher probability of occurrence. Along with the fact that most of the clusters stop propagating only when M_{L} is small, the computed results suggest that for an actively growing cluster to stop propagating the number of active drops should reduce, which makes this event less likely. This helps us understand why avalanches of a certain size range have lower probabilities of occurrence.

Hence, one can conclude that if a cluster does not stop propagating when its size is small, it is likely to grow to cover almost the whole system (see Fig. 1a and d). It is important that we understand the reason for this ‘runaway’ like behaviour. We propose that as the cluster grows, the size of Ac increases, which in turn amplifies the rate of cluster growth making the coalescence propagation autocatalytic. A plot of the average avalanche size as a function of the number of events (Fig. 3d) shows the sigmoidal nature of the growth which is a characteristic of an autocatalytic process. For example, Ritacco and co-workers also observed a sigmoidal like death curve when investigating the lifetime of bubble rafts.^{30} The authors use an algorithm where there is multi-drop interaction and report P_{A}(Av) with features similar to ours. However, they attribute their observation to the cooperative nature of the process where the death of a bubble influences the probability of the death of its neighbours. But in our simulations the drops do not show cooperative behaviour – because the probability of propagation is a function only of the orientation of drops – yet they show a hump in the P_{A}(Av) plot and a sigmoidal growth curve. As a result, we hypothesize that the sigmoidal like behaviour can only be a result of the autocatalytic nature of the propagation.

In an effort to understand the autocatalytic nature of the propagation, we study the number of neighbours available for coalescence using a simple conceptual measure that captures the nature of propagation. It is evident that during propagation each active drop has a certain number of neighbours n_{i} with which it can coalesce. Based on this, one could calculate an average number of drops around an active drop. However, this measure might not be very illuminating as the probabilistic nature of the propagation and the θ dependence of the probability does not favour coalescence through all the neighbours equally. Hence, we find the effective number of neighbours (n_{e}) available for coalescence by carrying out a weighted sum of the neighbours n_{e} = ∑P(θ_{i})n_{i}, where the weights are the associated propagation probabilities for the corresponding neighbours. We then use the idea of “number averaging” to find the average effective number of neighbours 〈n_{e}〉 averaged over time and all possible realizations, which is given by the expression . Remarkably, we find that the critical value where the transition to dominant autocatalytic behaviour occurs is at 〈n_{e}〉 = 3. This is a result that is quite non-obvious.

- P. J. a. Janssen and P. D. Anderson, Macromol. Mater. Eng., 2011, 296, 238–248 CrossRef CAS.
- L. G. Leal, Phys. Fluids, 2004, 16, 1833 CrossRef CAS.
- H. Gu, M. H. G. Duits and F. Mugele, Int. J. Mol. Sci., 2011, 12, 2572–2597 CrossRef CAS PubMed.
- Y. Yoon, M. Borrell, C. C. Park and L. G. Leal, J. Fluid Mech., 2005, 525, 355–379 CrossRef CAS.
- J. W. Ha, Y. Yoon and L. G. Leal, Phys. Fluids, 2003, 15, 849 CrossRef CAS.
- J. Eggers, J. R. Lister and H. A. Stone, J. Fluid Mech., 1999, 401, 293–310 CrossRef CAS.
- M. Borrell, Y. Yoon and L. G. Leal, Phys. Fluids, 2004, 16, 3945–3954 CrossRef CAS.
- D. Chen, R. Cardinaels and P. Moldenaers, Langmuir, 2009, 25, 12885–12893 CrossRef CAS PubMed.
- I. U. Vakarelski, R. Manica, X. Tang, S. J. O’Shea, G. W. Stevens, F. Grieser, R. R. Dagastine and D. Y. C. Chan, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 11177–11182 CrossRef CAS PubMed.
- N. Bremond, A. R. Thiam and J. Bibette, Phys. Rev. Lett., 2008, 100, 024501 CrossRef PubMed.
- T. Krebs, K. Schroën and R. Boom, Soft Matter, 2012, 8, 10650 RSC.
- K. Wang, Y. Lu, L. Yang and G. Luo, AIChE J., 2013, 59, 643–649 CrossRef CAS.
- G. F. Christopher, J. Bergstein, N. B. End, M. Poon, C. Nguyen and S. L. Anna, Lab Chip, 2009, 9, 1102–1109 RSC.
- L. Mazutis and A. D. Griffiths, Lab Chip, 2012, 12, 1800 RSC.
- L. Mazutis, J.-C. Baret and A. D. Griffiths, Lab Chip, 2009, 9, 2665–2672 RSC.
- J. C. Baret, F. Kleinschmidt, A. E. Harrak and A. D. Griffiths, Langmuir, 2009, 25, 6088–6093 CrossRef CAS PubMed.
- D. Y. C. Chan, E. Klaseboer and R. Manica, Soft Matter, 2009, 6, 20 Search PubMed.
- K. B. Migler, Phys. Rev. Lett., 2001, 86, 1023–1026 CrossRef CAS PubMed.
- N. Bremond, H. Doméjean and J. Bibette, Phys. Rev. Lett., 2011, 106, 214502 CrossRef PubMed.
- G. W. Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems and Adaptation, The MIT Press, 1999 Search PubMed.
- T. Beatus, T. Tlusty and R. Bar-Ziv, Nat. Phys., 2006, 2, 743–748 Search PubMed.
- B. M. Jose and T. Cubaud, Microfluid. Nanofluid., 2012, 12, 687–696 CrossRef CAS.
- M. J. Fuerstman, P. Garstecki and G. M. Whitesides, Science, 2007, 315, 828–832 CrossRef CAS PubMed.
- M. Schindler and A. Ajdari, Phys. Rev. Lett., 2008, 100, 044501 CrossRef PubMed.
- T. Beatus, R. H. Bar-Ziv and T. Tlusty, Phys. Rep., 2012, 516, 103–145 CrossRef.
- N. Desreumaux, J. B. Caussin, R. Jeanneret, E. Lauga and D. Bartolo, Phys. Rev. Lett., 2013, 111, 118301 CrossRef PubMed.
- M. Danny Raj and R. Rengaswamy, Microfluid. Nanofluid., 2014, 17, 527–537 CrossRef CAS.
- D. Y. C. Chan, E. Klaseboer and R. Manica, Soft Matter, 2011, 7, 2235 RSC.
- A. Lai, N. Bremond and H. a. Stone, J. Fluid Mech., 2009, 632, 97 CrossRef.
- H. Ritacco, F. Kiefer and D. Langevin, Phys. Rev. Lett., 2007, 98, 244501 CrossRef PubMed.

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