Heat bath coupling

It is the formal framework of thermodynamics that provides the possibility of coupling to a certain heat bath. It is given by the first law of thermodynamics, where thermal heat exchangeδQis provided by Eq. (2.37). It allows for controlling both the initial temperature quench and the equilibration during the demixing process. In SPH, as described in Sec. 3.1.6, it is not necessary to treat simulations adiabatic (whereδQ= 0), but also to model on this basis a heat exchange between the particles. Moreover, thermal exchange with an external heat source is also feasible. These so called thermostats are common methods to treat temperature realistic in microscopic simulation methods, such as they are used by molecular dynamics (MD) or Monte Carlo [70]. The transfer of thermostats from MD to SPH is obvious, since both methods are very similar [71].

Fig. 2.1 gives a schematic representation of the evolution in the phase diagram of spinodal decomposition following a certain temperature quench. The stage of decomposition in this figure is shown isothermally, which is in accordance with available theoretical predictions (for example the well-known Cahn-Hilliard equa-tion). However, in general, this is a non-isothermal process and can not withstand a comparison with real phase separating fluids, where temperature locally devi-ates from the mean system temperature. Thus, for this purpose hydrodynamics must be taken into account. Moreover, considering the vdW-SPH method, it can be seen from Eq. (3.59) that the particle temperatures are expected to increase or decrease with the respective density deviation. It can be declared as cooling and heating that is due to expansion and compression of the fluid, respectively.

Thus, depending on the actual initial quench conditions and the corresponding mass fraction, the equation is an explanation for the release of latent heat, if the initial density is chosen large enough. Note that the contrary decreasing mean temperature can hardly be realized for a liquid-vapor system. This is because of the initial density that is needed to be chosen at a very low value in order to obtain an adequate mass fraction, whereby it would transit the spinodal curve and enter the classical nucleation regime. Thus, no spinodal decomposition will be observed.

Release of latent heat needs to be compensated for a comparison with theoreti-cal predictions. Note that the original SPH method, as described above, conserves total energy, where an increase of kinetic energy of the fluid is accompanied with a loss of total internal energy and vice versa. This is a direct consequence of the hydrodynamic equations. Moreover, taking artificial dissipation and thermal conductivity into account, where the dynamic process becomes irreversible, conser-vation of total energy is still given. This immediately changes when a thermostat is applied for the temperature tuning.

In the following the two realizations of thermostats that have been used for this work will be described. The first thermostat instantaneously scales the mean tem-perature to a predicted value. This type of thermostat can be used to explain the domain growth behavior of pure non-isothermal systems where the temperature increases during decomposition. This can be done by assigning the growth rates

obtained at different quench depths to certain regimes in the pure thermal system.

The second is the time-scale thermostat, where each local particle temperature ex-ponentially decays to the beforehand predicted temperature at a prescribed time scale. By varying the time scale, this is a model for the strength of coupling to an external heat bath. The time-scale thermostat is used to describe a transition from weak to strong coupling between the limits of pure thermal simulations and instantaneously temperature scaled systems.

Scaling thermostat

As already mentioned, it is due to the caloric vdW-EOS (3.59) that the mean temperature T of the system may increase during the phase separation process.

If the focus of interest is on the evolution of these separated phases, it might be desirable to keepT at a constant desired value T0. One way would be to keep all particle temperaturesT_i fixed atT₀, which would generate an isothermal process, but could not preserve thermal fluctuations within the dynamical system. The two remaining possibilities are either shifting or scaling the mean temperature to the desired value. The former is not recommended because particle tempera-tures below T can become negative. Hence, the temperature scaling approach is implemented in the following way. The new particle temperatures ˜Ti are given by

Te_i=T_i·T₀

T . (3.62)

This procedure will be herein further on referred to as the scaling thermostat. It should be noticed that in molecular dynamics (MD), thermostat methods, espe-cially the velocity rescaling method, are used in a very similar way to realize a constant temperature ensemble [70]¹³.

Actually, the SPH method in its simplest non-adaptive formulation is closely related to MD [71]. The velocity rescaling method for MD does not reflect the proper physical properties of a canonical ensemble and must be extended to more advanced but also common methods (such as the Berendsen coupling or Nos´ e-Hoover thermostat), because it does not allow for local fluctuations of the tem-perature. However, for SPH the scaling approach is quite suitable, because it already treats the temperature as a mesoscopic field quantity. Furthermore, it allows the temperature field to fluctuate around a given mean value.

Timescale thermostat

It is apparent that the instantaneous temperature scaling approach is not a realistic model, because it does not take the strength of coupling into account. A somewhat more realistic model is given by the time-scale thermostat. It is inspired by the

13In statistical physics, this type of ensemble is commonly referred to as the canonical ensemble, orN V T-ensemble, which generally indicates a system with constant particle number, volume, and temperature values

Chapter 3 Algorithms and methods

so called Berendsen thermostat [72], which is a very common choice for molecular dynamics simulations. The thermal evolution of the mean temperature is given by the solution of the ordinary differential equation

dT dt = 1

τ(T0−T), (3.63)

where T is the arithmetic mean, τ is a constant that defines the thermalization time scale andT0 is the desired final mean temperature. For the calculation ofT, Eq. (2.9) is used to assign a local temperature to each SPH particle. The solution of Eq. (3.63) depends on the integration timestep ∆tand is given by

T_new= (T_old−T₀)e^−∆tτ +T₀, (3.64) where ∆t = maxi∆ti and Tnew(t) = T_old(t+ ∆t). Note that a more detailed discussion on integration schemes and conditions on timestep size will be given in Sec. 3.3.1. The new temperatureTe_i of a particleiis then obtained by

Te_i =T_i

T_new Told

. (3.65)

It is clear that for the limitτ →0, Eq. (3.65) and (3.64) are converge to Eq. (3.62).

In contrast, when τ → ∞, it corresponds to a pure thermal simulation where no thermostat is applied.

Note that for molecular dynamics (MD) the Berendsen thermostat is known to suppress thermal fluctuations, which is true for microscopic simulation methods, but for our mesoscopic scheme, where only the overall mean temperature is scaled by the exponential factor to the desired value, thermal fluctuations are conserved.

If latent heat is released, then τ defines the half time of the process where the heat is conducted to the heat bath.

Furthermore, the timescale thermostat provides an excellent opportunity for modeling the initial quench, if the quench is desired to become part of the sim-ulation. Most important reasons for this procedure is the comparison between simulation, theory and experiment, where theoretical predictions mostly consider instantaneous quenches. This is in contrast to experiments, where the quench at finite quench rate is always part of the experiment and it cannot be considered separated from the subsequent demixing process.