In this paper, we analyze more than 16 million tweets that have been sent from 6.1 million Twitter accounts and are related to nine natural disasters. As part of our analysis, we identify eight basic emotions conveyed in these tweets. We found that during natural disasters, social media messages at first predominantly express fear, while sadness and positive emotions increase in the immediate aftermath of the event. In this context, positive emotions contribute to the social phenomenon of emotional bonding and are often related to compassion, gratitude, as well as donations for disaster relief. In our analysis, we found that the users’ emotional expressions directly contribute to the emergence of the underlying communication network. In particular, we identified statistically significant structural patterns that we call emotion-exchange motifs and show that: (1) the motifs 021U and 021D are common for the communication of all eight emotions considered in this study, (2) motifs which include bidirectional edges (i.e. online conversations) are generally not characteristic for the communication of surprise, sadness, and disgust, (3) the structural analysis of a set of emotions (rather than a single emotion) leads to the formation of more complex motifs representing more complex social interactions, and (4) the messaging patterns emerging from the communication of joy and sadness show the highest structural similarity, even reaching a perfect similarity score at some point during the data-extraction period.