Clouds act as thermostats for the atmosphere; controlling global temperatures through the hydrological cycle and their interaction with radiation. Therefore, understanding the formation and development of clouds is essential in determining the possible future effects of climate change. One of the most uncertain components of climate modelling is the representation of cirrus clouds within general circulation models. The two key challenges limiting the accuracy of cirrus cloud modelling are our ability to make observations in the upper troposphere and our knowledge of the micro-physical processes that lead to ice nucleation at low temperatures. Observations of cirrus clouds in the upper troposphere, although limited, have found higher than expected supersaturations and lower than expected ice crystal numbers, which suggests that our current understanding of the ice formation processes under these conditions are incomplete. A number of different theories have been proposed to explain the consistently high relative humidities, one of these is the role of highly viscous secondary organic aerosols in ice nucleation at low temperatures high in the troposphere. This thesis focuses on the development of a numerical model to simulate the effect of viscosity on ice nucleation in low temperature cirrus. Through the introduction and literature review; a motivation and scientific basis for the study are established in order to develop the project's aims and objectives. The novel research is then contained within the following three chapters; the first describes the development of a numerical framework to model condensed phase diffusion and phase separation within individual aerosol particles; the second, establishes a new modelling approach to investigate whether ice formation within clouds is sensitive to aerosol viscosity; and the final study furthers this work to focus on the formation of low temperature cirrus clouds high the troposphere near to the tropical tropopause. The initial development of a numerical diffusion framework shows that viscous effects of diffusion dominate the rate of mixing at low relative humidities, but at high relative humidities, mixing timescales were more sensitive to solubility effects. By including non-ideality through the activity coefficients in the Maxwell-Stefan equation, the simple single particle diffusion framework is able to reproduce liquid-liquid phase separations that have been observed in a number of laboratory experiments. The second study finds that the rate of condensed phase diffusion significantly affects the rate of droplet growth within a cloud, which under particular conditions of temperature and updraft velocity can alter the number of ice crystals formed in model simulations. Here, simple constant diffusion coefficients, which are unrealistic under atmospheric conditions, are used to investigate model sensitivities and allow for model development. Hence, the final study uses diffusion coefficients for secondary organic aerosol, evaluated from laboratory experiments, to show that there is a complex relationship between temperature, updraft velocity and ice nucleation rates in low temperature cirrus clouds. Suggesting that by taking into account aerosol viscosity, the formation of cirrus clouds could be better replicated in numerical models. Through numerical modelling techniques, the findings in this thesis contribute towards a better understanding of how viscous aerosol interact with water vapour in the atmosphere and how the formation of low temperature cirrus clouds is affected, which has important implications for determining the future of the changing global climate.