We present an analysis of the early, rising light curves of 18 Type Ia supernovae (SNe Ia) discovered by the Palomar Transient Factory and the La Silla-QUEST variability survey. We fit these early data flux using a simple power law (f(t) = alpha x t...) to determine the time of first light (t...), and hence the rise time (t...) from first light to peak luminosity, and the exponent of the power-law rise (n). We find a mean uncorrected rise time of 18.98 plus or minus 0.54 d, with individual supernova (SN) rise times ranging from 15.98 to 24.7 d. The exponent n shows significant departures from the simple 'fireball model' of n = 2 (or f(t) ... t...) usually assumed in the literature. With a mean value of n = 2.44 plus or minus 0.13, our data also show significant diversity from event to event. This deviation has implications for the distribution of ...Ni throughout the SN ejecta, with a higher index suggesting a lesser degree of ...Ni mixing. The range of n found also confirms that the ...Ni distribution is not standard throughout the population of SNe Ia, in agreement with earlier work measuring such abundances through spectral modelling. We also show that the duration of the very early light curve, before the luminosity has reached half of its maximal value, does not correlate with the light-curve shape or stretch used to standardize SNe Ia in cosmological applications. This has implications for the cosmological fitting of SN Ia light curves. (ProQuest: ... denotes formulae/symbols omitted.)