In case you are interested at the glass transition, which has been thought as somehow resembling a second order phase transition in terms of the Ehrenfest classification, even if this analogy is based at idealized circumstances that can not be realized in practice, you may check:

Article Aspects of the non-crystalline state

Thanks for your kindly recommendation, Mr Queiroz. It seems the transition from liquid to glass has no fixed transition temperature and cannot be simply illustrated by mean field  theory. So I am still confused why many people thinks glass transition is a second order transition. I will read your paper, thanks.

I think you can find the answer to your question yourself. The temperature change with time dT/dt is determined by the heat influx dQ/dt =q into your sample according to dQ/dt=C_p dT/dt. In the case of a first order transition C_P is infinite at the transition temperature and thus dT/dt=0 for the time t=Q_latent/q it takes to let the latent heat flow. For a second order transition the same equation should apply. Let's work with dQ/dt=const.=q. In order to get T(t) you have to solve the first order differential equation dT/dt=q/C_P(T). This can be done via separation of variables. Solve Integrate[C_P(T'),{T',T_i,T}]=q t . Work with the diverging part of the heat capacity which, for a second order transition, reads C_P=C*Abs[T-T_c]^(-alpha). The result depends of course on alpha. The integral has to be taken in two steps. First integrate to T_c-epsilon and then from T_C+epsilon to final T. The point is that despite the fact that the heat capacity diverges I expect the integral to be finite and the limit epsilon->0 should not create any infinities. This is why T(t) should be continuous. The specific result depends on the critical exponent alpha of the transition. This is how I see it at least.