We use computer simulations to study the relaxation dynamics of a model for oil-in-water microemulsion droplets linked with telechelic polymers. This system exhibits both gel and glass phases and we show that the competition between these two arrest mechanisms can result in a complex, three-step decay of the time correlation functions, controlled by two different localization lengthscales. For certain combinations of the parameters, this competition gives rise to an anomalous logarithmic decay of the correlation functions and a subdiffusive particle motion, which can be understood as a simple crossover effect between the two relaxation processes. We establish a simple criterion for this logarithmic decay to be observed. We also find a further logarithmically slow relaxation related to the relaxation of floppy clusters of particles in a crowded environment, in agreement with recent findings in other models for dense chemical gels. Finally, we characterize how the competition of gel and glass arrest mechanisms affects the dynamical heterogeneities and show that for certain combination of parameters these heterogeneities can be unusually large. By measuring the four-point dynamical susceptibility, we probe the cooperativity of the motion and find that with increasing coupling this cooperativity shows a maximum before it decreases again, indicating the change in the nature of the relaxation dynamics. Our results suggest that compressing gels to large densities produces novel arrested phases that have a new and complex dynamics.

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