RESULTADOS Y DISCUSIÓN

The goal of the machine learning approach is to distinguish, based purely on structure, those particles that are likely to rearrange from those that will be stable for a long time. To this

Figure 5.2: Poly(N-isopropyl acrylamide) (PNIPAM) particle trajectory and its correspondingPHop

values. (a)A typical trajectory of a particle in the xy-plane that undergoes a large rearrangement. The x- and y-axes (horizontal and vertical, respectively) are scaled to the large particle diameter, σ1. The inset shows a cropped image of the PNIPAM particles. Large particles appear brighter.

(b)ThePHop trajectory of the same particle from(a). Because this particle’s peak value ofPHop is

greater thanPH, this particle is included in the training set class of “rearranging” particles for the

end we employ an SVM [262, 263]. Briefly, we desire structural features that can represent the local caging structure around particles but which are also capable of distinguishing the classes of rearranging and stable particles.

As a hypothetical example, consider two features, F1 and F2, which are computed for every particle. If we compute these features for particles known a priori to be rearranging or to be stable, and if we label them as such in the 2-dimensional feature space, then the goal of our algorithm is to show that the labeled groups of particles are well separated as in Fig. 5.3. These two labeled groups are collectively called the SVM’s “training set”. In other words, we seek the line (in the 2-dimensional hyperspace) that best separates rearranging from stable particles in the training set. The fraction of particles in the training set that are correctly divided by the hyperplane is the SVM’s accuracy.

In practice, having selected the particles that make up the training set, we train two SVMs, one for each particle species using generic local 2-point structural features that represent the cages surrounding each (i-th) particle [264]. The functions we choose are radial correlation functions:

GX_Y (i;µ) =X

j6=i

e−(Rij−µ)2/l_{, (5.2.2)}

where j runs over all the particles within 5σ1 radius, X and Y indicate the species of the

i-th and j-th particle respectively, l = 0.1σ1, and µ takes all values between 0.3 and 5.0 in increments of 0.1. Thus there are for each neighboring species 47 features that describe the local cage environment of each particle for a total of 94 “structure features”. Note, the features we chose only characterize the radial distribution of the cage forming neighboring particles We also tested features that characterize bond angle between neighbors, but we did not find them to improve the accuracy of the SVM hyperplane. For the rearranging class of particles, the features are computed at a time-delay (τp) prior to when the particle’s

PHop value increases above PL. For stable particles, the features are computed in the first

Figure 5.3: Description of support vector machine (SVM) method. Particles are first classified as “rearranging” (blue crosses) or “stable” (green dots) by e.g. PHop (see text). Then features that

represent their local caging are computed. In this case, the two hypothetical features,F1andF2are

found to well separate the classes of rearranging and stable particles (though not perfectly). The SVM accuracy is the fraction of particles correctly sorted by the hyperplane. The SVM algorithm generates a hyperplane (red dotted line) that best separates the two classes of particles. A measure- ment of a new particle’s (black square) featuresF1andF2then can be used to predict if that particle

will rearrange or be stable. We define the particles “softness” as its signed distance (displacement) from the hyperplane (doubled headed arrow).

the sample, we only utilize particles that are at least 5σ1 away from the edge of the field of view in the training set and in the results computed below. The SVM training accuracy for large particles and small particles is 85% and 80% respectively.

The trained SVM hyperplane is then employed to characterize the entirety of the ob- served data (the “test” sample). This process yields softness values of the whole system [58], i.e., for every particle at every instant in time. Briefly, the result of the SVM train- ing is a 93 dimensional hyperplane that best separates the cage structures of rearranging particles from the cage structures of the most stable particles. The cage structures of all other particles in our experiments (the “test” samples) are then computed at all times using the structure features of equation 5.2.2. Finally, the signed distance (displacement) from the hyperplane is then computed for each particle at each time point. This signed distance (displacement) is called the “softness”; positive values of softness indicate a particle is more likely to rearrange, and more negative values indicate a particle is more stable.

Once an SVM is trained on a sufficiently large dataset, the softness parameter can be determined from a purely structural measurement such as an image of the particle ensemble. Given a single image frame, the same hyperplane can be used to compute softness of particles in other colloidal glasses with similar packing conditions and interactions. The goal then is to employ softness, a structural measure of particle packings, to predict dynamics of the particles and ultimately to characterize the properties of the bulk colloidal glass. In the following section, we show that softness is predictive of the residence times that particles will spend in their cages, and thus is predictive of the local activation energy barriers they must overcome to rearrange.