Billions of IoT devices are deployed worldwide and batteries are their main power source. However, these batteries are bulky, short-lived and full of hazardous chemicals that damage our environment. Relying on batteries is not a sustainable solution for the future IoT. As an alternative, battery-less devices run on long-lived capacitors charged using energy harvesters. The small energy storage capacity of capacitors results in an intermittent on-off behaviour. LoRaWAN is a popular Low Power Wide Area Network technology used in many IoT devices and can be used in these new scenarios. In this work, we present a Markov model to characterize the performance of battery-less LoRaWAN devices for uplink and downlink transmissions and we evaluate their performance in terms of the parameters that define the model (i.e., device configuration, application behaviour and environmental conditions). Results show that LoRaWAN battery-less communications are feasible if choosing the proper configuration (i.e., capacitor size, turn-on voltage threshold) for different application behaviour (i.e., transmission interval, UL/DL packet sizes) and environmental conditions (i.e., energy harvesting rate). Since downlink in the second reception window highly affects the performance, only small DL packet sizes should be considered for these devices. Besides, a 47 mF capacitor can support 1 Byte SF7 transmissions every 60 s at an energy harvesting rate of 1 mW. However, if no DL is expected, a 4.7 mF capacitor could support 1 Byte SF7 transmissions every 9 s.