Nucleation. Thermodynamical and Stochastical Descriptions

Nucleation. Thermodynamical and Stochastical Descriptions A thesis submitted by Antonio Patr´on Castro Supervised by Manuel Morillo Buz´on Department of Atomic, Molecular and Nuclear Physics University of Seville June, 2019 “The most merciful thing in the world, I think, is the inability of the human mind to correlate all its contents... some day the piecing together of dissociated knowledge will open up such terrifying vistas of reality, and of our frightful position therein, that we shall either go mad from the revelation or flee from the light into the peace and safety of a new dark age.” -H.P. Lovecraft [11] Contents 1 Abstract 1 2 First-Order Phase Transitions 2 2.1 VanderWaalsFluid ............................... 2 2.2 Thermodynamic Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Classical Nucleation Theory 7 3.1 Thermodynamics of Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.1 The Capillary Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.1.2 Work for Cluster Formation . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Kinetic Theory of Nucleation . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.1 Nucleation Stage and the BDT Model . . . . . . . . . . . . . . . . . 15 3.2.2 Equilibrium Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2.3 Nucleation Rate in CNT . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 Homogeneous Nucleation of Water . . . . . . . . . . . . . . . . . . . . . . . 22 3.3.1 Equilibrium Distribution and Critical Cluster . . . . . . . . . . . . . 23 3.3.2 The Cloud Chamber and Temperature Dependencies . . . . . . . . . 26 3.4 DeficienciesoftheCNT ............................. 28 4 Density Functional Theory 30 4.1 Density Functional and Variational Approach . . . . . . . . . . . . . . . . . 31 4.2 Approximations for the Free Energy . . . . . . . . . . . . . . . . . . . . . . . 33 4.2.1 Gradient Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2.2 Hard-SphereModel............................ 34 4.3 N-ParameterTheories .............................. 35 5 Conclusion 36 1 Abstract Nucleation is a non-equilibrium process through which a system evolves from an old thermodynamic phase to a new (energetically stable) one in what we call a first-order phase transition. This process is widely spread in natural phenomena like condensation, sublimation, evaporation or crystal growth, and it is also of technological relevance [1]. In this work, the main theories behind nucleation developed throughout the last century by researchers like Gibbs, Zeldovich and Kashchiev will be reviewed. We will start with the study of the Van der Waals equation (VDWE), as it provides a general insight of the main conditions required for a phase transition to occur. Next, the Classical Nucleation Theory (CNT), the core of this work, with its central paradigm of well-defined clusters is introduced. With the aid of the capillary model and the assumption of large spherical clusters, a first approach to estimate the minimum work required for a cluster to form (also known as the potential or nucleation barrier) is obtained, with a general description of how the cluster grows depending on its initial size. Moreover, the CNT also provides a dynamical theory of cluster formation based on the attachment and detachment of n-mers (clusters formed by ’n’ molecules), which leads to Fokker-Planck, Langevin, Kramers-Moyal and Master equations, due to the stochastic nature of the time-dependent concentration of n-mers in the system. With the help of the BDT (Becker-D¨oring-Tunitskii) model and the Zeldovich’s ideas, an analytical expression for the nucleation rate is derived, which constitutes the main quantity that allows to corroborate the theoretical predictions experimentally. With this framework, we will focus on the study of homogeneous nucleation of water droplets in condensate vapor phases, with the goal of obtaining the cluster size equilibrium distribution functions and the performance of a comparison between the theoretical and experimental nucleation rates measured in diffusion cloud chambers. This procedure will allow us to study the limitations and validity of the classical theory, and its study will conclude with a section where an oversight of its main controversies and deficiencies is provided. Finally, following this, we will make a brief introduction presenting the framework and some fundamental approximations of the modern paradigm of the Density Functional Theory, which is based on variational calculus. This theory provides a completely new framework that improves the classical theory and opens the door to new models that, in concordance with the experiments, allows us to deepen in this field of study, helping us to uncover the mysteries of phase transitions and matter itself. 1 2 First-Order Phase Transitions Let us consider a macroscopic system whose properties and time evolution will be studied. This system might have different thermodynamic phases. We define a phase as a spatial region of the system whose macroscopic properties remain homogeneous for a given set of variables, due to particular arrangements of its atoms and molecules and their interactions. Many instances of systems showing different phases are found in Nature. A most typical case is water, which can be found in three different phases (liquid, solid or gas) depending on the temperature and pressure. Other examples are ferromagnetic materials, which can present spontaneous magnetisation depending on its temperature, or crystalline solids, whose atoms may be arranged in different lattices depending again on the external parameters. A phase transition, from a thermodynamic point of view, is defined then as the process through which the system changes its macroscopic properties from the ones corresponding to the old phase to the new ones [1]. The first classification of possible types of phase transitions was proposed initially by Paul Ehrenfest (1933) [1]. According to Ehrenfest’s proposal, phase transitions are classified depending on the behaviour of the Gibbs free energy at the transition point. It is said that we have an n-order phase transition if the n-th derivative of the Gibbs free energy at the transition point presents a discontinuity. Further experiments and investigations carried out by Onsager (1944), Gebhardt (1980) and others [1] showed that this classification does not include all possible cases. Here, we will focus on the study of first-order phase transitions, characterised by discontinuities of the internal energy U, the entropy Sand the volume V of the system. In order to illustrate the physical nature of first-order phase transitions, it is worth to analyze the behavior of a Van der Waals (VDW) fluid, as we do next. 2.1 Van der Waals Fluid A VDW homogeneous fluid is characterised by the following equation of state p+N2a0 V2(V−Nb0) = NkBT, (2.1) where pis the pressure, Vthe volume, Nthe number of particles, Tthe absolute temperature, kBthe Boltzmann’s constant, and a’ and b’ are constants that account for the attraction between particles and their effective volume respectively. They depend on the material under study [1]. It is interesting to outline that this equation can be derived 2 from statistical-mechanical considerations, assuming a Hamiltonian including a dominant hard-sphere potential and a Lennard-Jones potential which is treated as a thermodynamic perturbation [6]. This allows us to visualize the connection between the material constants and the microscopic interactions between molecules. Moreover, these constants may be expressed in terms of the critical coordinates, which are obtained from the definition of a saddle point as ∂p ∂V Tc =∂2p ∂V 2Tc = 0. These equations lead to pc=a0 27b02, Vc= 3Nb0, Tc=8a0 27kBb0. By substituting the reduced coordinates p0=p/pc,V0=V/Vcand T0=T/Tcinto Eq.(2.1) we obtain the reduced VDW equation p0+3 V02(3V0−1) = 8T0.(2.2) This last expression is universal as it does not depend on any particular material constants. Let us now discuss the thermodynamic potentials governing the behaviour of our fluid. Considering that our system is placed within a thermal bath of temperature TBand constant pressure pB, the fluid will be in stable thermodynamic equilibrium (TDE) when it occupies a volume for which the Gibbs free energy G=F−pBVis minimum [7]. Knowing that the differential expression for the Helmholtz free energy Fis given by dF =−SdT −pdV , we can obtain the conditions for obtaining a minimum of the Gibbs free energy, ∂G ∂V T =∂F ∂V T +pB=−p+pB= 0 →p=pB, ∂2G ∂V 2T =∂2F ∂V 2T =−∂p ∂V T >0→∂p ∂V T <0. On one hand, the first condition assures us that we can reach the stable TDE if our system’s pressure is equal to the external pressure. This result makes physical sense, because a difference (gradient) of pressures would result in a force which would try to expand or contract our system until this difference vanishes. On the other hand, the second condition assures that the pressure of the fluid decreases if we expand its volume, a well known fact to be true for the majority of common fluids. 3 (a) 1a (b) 1b Figure 1: Reduced p-V diagrams for VDW fluid. Fig.(1a) shows the critical isotherm (black) and a T’= 0.85 isotherm with p0 B= 0.505 [1], where both blue regions have the same area, according to Maxwell’s construction. Fig.(1b) [10] shows several isotherms in order to visualize the binodal (continuous line joining a, c and C points) and spinodal (dashed line joining d, e and C points). Figure 1 shows two similar reduced p-V diagrams for different isotherms. On one hand, isotherms of temperature greater than the critical one Tcare known as supercriticals, where there is no distinction anymore between the liquid or gas phases of the fluid. Moreover, for even greater temperatures, the VDW fluid can be regarded as an ideal gas. On the other hand, those isotherms whose temperature is lower than the critical one are known as subcriticals. We will be interested in those isotherms, where there is a clear distinction between the liquid and gas phases of the fluid and thus, we can talk about a phase transition between them. There are two key concepts that emerge by applying the TDE conditions for a subcritical isotherm [1]: •Spinodal Curve: As it can be noticed in both graphs from Figure 1, for a subcritical isotherm, there is a region between the p’(V’) local minimum and local maximum where the second stability condition breaks down. Namely, the region is unstable. As we increase the temperature, this region shrinks until the maximum and minimum coincide at the critical point. The curve obtained by joining local extremes for different subcritical isotherms is called spinodal. 4 •Binodal Curve: For a given external pressure p0 Bat a subcritical temperature, Eq.(2.2) presents three solutions for the volume V, two of which are of physical interest while the third one lies within the spinodal region, which is unstable. They correspond to a liquid or gas in stable TDE. When both solutions correspond to the same minimum of the Gibbs free energy, then the liquid and gas are in phase equilibrium, that is, both phases coexist as separate phases at what is called the equilibrium pressure p0 B=p0 E1of the fluid. By joining the coexistence points for different subcritical temperatures, the binodal curve is obtained. Its maximum is located at the critical point, where both phases coexist, but their boundaries vanish, so there is no distinction anymore between both phases and then we have what is known as a supercritical fluid. Now, the most interesting case of this section lies when we have an external pressure p0 B6=p0 E, and between the p’(V’) local maximum and minimum pressures. In this case, both liquid and gas phases would be in TDE, but corresponding to minima of the Gibbs free energy of different depth, that is, since the deepest one it is said to be in truly stable TDE, the second one corresponds to a metastable TDE. The metastable states lie within the two regions (metastable liquid and metastable gas) between the binodal and spinodal curves, and the transition from a metastable state to the truly stable state corresponds to a first-order phase transition. The process through which this transition takes place is the nucleation of the old (metastable) phase into the new (truly stable) one [1]. It is important to outline that the phase transition depends on the difference between the corresponding truly stable and metastable free energies ∆g[1], or more explicit, the quotient S=p0 B/p0 E2[2]. Both quantities, for the case of nucleation of liquid droplets in condensed gas phases, are related by means of the Kelvin formula [1] ∆g=−kBT v0 log S, (2.3) where v0is the volume of a molecule from the system. The quantity Sis known as the thermodynamic driving force or supersaturation. Physically, it represents the gain in free energy per particle associated with the transition from the old to the new phase, and it establishes both the direction and the strength of the transition. For ∆g= 0 (or p0 B/p0 E= 1), there exists no preferred direction nor driving force for the transition, and hence, for 1The equilibrium pressure p0 Emay be obtained by applying Maxwell’s rule; the pressure for which the blue areas on Figure 1a have the same area. These areas represent the excess heat gained and transferred by the system in its metastable states, and whose balance must be null, so the total amount of heat involved in the transition is p0 E(V0 f−V0 i) [7]. 2For sufficient diluted (weak) solutions and ideal gases, this ratio is approximately equal to the ratio of densities of the two states (metastable and truly stable), that is: p0 B/p0 E≈ρmet/ρE[2] 5 nucleation, and then the old phase is said to be saturated, while for ∆g > 0, the transition is not possible in the desired direction. In this case the old phase is said to be undersaturated [1]. 2.2 Thermodynamic Fluctuations We have determined the necessary conditions for a phase transition to occur, but our analysis does not include the time needed for the transition to happen. Indeed, the supersaturated phase will remain a certain time on its metastable state due to the fact that there is an energy barrier between both TDE states. Moreover, this energy barrier will have a maximum of different heights depending on the path followed during the transition. Then our next problem would consist in determining the path through which the energy cost is minimized. One could think of a path corresponding to the uniform change of density from the old phase to the new one as the desired one. However, this path is not actually followed due to its high energy cost. Another possibility proposed by Kashchiev [1], energetically much cheaper, consists in a non-uniform change of density due to themodynamic fluctuations on the density. In this case, the energy barrier would be much lower due to the fact that in this case the number of particles involved in the transition is much less than in the uniform case, where all the particles would participate in the process. This approach can describe multiple everyday phenomena such as rains and snowfalls, where droplets and flakes are just the manifestation of density fluctuations. By following this path, nucleation can be defined, according to Kashchiev, as the process of random generation of nanoscopically small formations of the new phase that have the ability for irreversible overgrowth to macroscopic sizes. [1] From Einstein’s theory of thermal fluctuation [7], the probability of occurrence of those density fluctuations is exponentially proportional to the minimum work required for the process to occur, P∝exp −Wmin kBT,(2.4) where Tis the absolute temperature and kBis the Boltzmann’s constant. The estimation of the minimum work Wmin is one of the main goals of both the Classical Nucleation Theory, developed mainly by Gibbs, and of the Density Functional Theory, based on variational calculus. This goal, together with the proposal of a kinetic model to describe the generation and time evolution of clusters, constitutes the core of every nucleation theory. 6 With these results, the thermodynamics of nucleation developed by Gibbs has been completely discussed. Its main goal consisted in obtaining the maximum work the system would need in order to create a critical nuclei, which we derived to be the maximum height of the energy barrier, also known as nucleation barrier. It reflects that the energy balance of the cluster within the old phase depends mainly on two terms; the volume term, which is the energy cost of forming an spatially small new phase of different density within the initial phase, and the surface term, which takes into account the energy cost for creating a dividing surface between the cluster and the old phase. In the following section we will obtain a kinetic constant that, together with the probability exponential provided by Eq.(3.13), allows us to formulate an analytical expression for the nucleation rate, which constitutes the second most important quantity to study on this work. 3.2 Kinetic Theory of Nucleation Up to this point, we have discussed the energies involving the formation of a cluster with similar properties as the new, truly stable, phase within the old metastable one. However, now we must ask ourselves how actually this formation process takes place, that is, the nucleation dynamics. In the context of the CNT, we will follow what is known as the Szilard-Farkas model, which is based on the following assumptions [1]: •Clusters are well defined entities formed by an arbitrary number of molecules n. This number constitutes the ’size’ of the cluster. It fully characterizes the nucleation process. Then we say that this model is based on a single order parameter theory. A cluster containing nmolecules is known as a n-mer. Thus, an unitary cluster would be a monomer, a cluster containing 2 molecules a dimer, then trimer and so on. •Clusters grow or shrink by the attachment or detachment of m-mers. This growing or decaying process is characterized by the transition frequencies fnm(t), that take into account the number of transitions n→mper unit time. Due to the fact that the clusters are fully determined by its size n, there is no need to describe molecular exchange between clusters of the same size. With these assumptions, we define Cn(t) as the concentration of n-mers at a given instant t [2]. The time evolution of this quantity will depend on the possible outgoing and incoming transitions. We will call fnm(t)Cn(t) the number of n→mtransitions per unit time and 13 volume. Thus, at a given instant of time t,Cn(t) will increase at the rate X m6=n fmn(t)Cm(t),(3.16) while it will decrease with rate X m6=n fnm(t)Cn(t),(3.17) It is then clear that the time evolution of Cn(t) will be given by dCn(t) dt =X m6=n [fmn(t)Cm(t)−fnm(t)Cn(t)] ,(3.18) which is known as the Master Equation of Nucleation (MEN) [2]. Actually, instead of a single equation, the MEN describes in principle, an infinite set of ordinary differential equations (ODEs). However, for practical and physical reasons, one typically sets a maximum size in order to get a finite set of ODEs. Eq.(3.18) is called the MEN because of the similarities with the master equations derived from Chapman-Kolmogorov equations in the discrete theory for Jump Markovian Processes [3]. However, this name is not completely well employed since the transition frequencies fnm(t) generally depend on the concentration Cn(t) of n-mers, making Eq.(3.18) a non-linear equation [2]. It is also interesting to develop a continuity equation from the MEN, which will be useful in the following sections. This can be carried out by defining the flux of clusters that grow from an arbitrary size m0≤nup to a size m > n, which can be expressed as jn,m(t) = n X m0=1 [fm0m(t)Cm0(t)−fmm0(t)Cm(t)] .(3.19) The net flux of clusters passing through the size nwill be jn(t) = Pm≥n+1 jn,m(t), that is, the addition of all the possible mfluxes. With these definitions, the MEN can be rewritten as dCn(t) dt =−(jn(t)−jn−1(t)) = −∆njn(t),(3.20) which has the form of a discrete continuity equation. Until now we have derived several expressions that imply summations over finite differences, due to the fact that the cluster size is defined as an integer number, for it represents the number of molecules in the cluster. However, in the context of the capillary model presented in the previous section, the order parameter employed was the cluster radius (considering spherical clusters, of course), which, in the end, is directly related to the cluster size n. In order to simplify the mathematics, 14 Zeldovich [1, 2] proposed that, if the radius is presented as a continuous variable, then the cluster size could be considered as a continuous variable too, so Eq.(3.18), (3.19) and (3.20) can be rewritten in terms of integrals as follows dC(n, t) dt =ZM 1 dm[f(m, n;t)C(m;t)−f(n, m;t)C(n;t)],(3.21) j(n, t) = ZM n dm ZM0 1 dm0[f(m0, m;t)C(m0;t)−f(m, m0;t)C(m;t)],(3.22) dC(n, t) dt =−∂j(n;t) ∂n .(3.23) The later expressions are only valid for large cluster sizes, for which the change between next neighbours clusters may be considered small enough. 3.2.1 Nucleation Stage and the BDT Model The MEN provides a general insight of how clusters evolve through time depending on the attachment and detachment rates of m-mers. However, Eq.(3.18) and Eq.(3.20) imply the solution of ordinary differential equations with finite series and integrals with unknown coefficients and functions fnm(t) and f(n, m;t) respectively. Further approximations are required in order to continue. By taking into account that we are concerned with the initial processes occurring during the early stage of the phase transition, that is, the nucleation stage, then it is highly unlikely for clusters greater than monomers (m= 2,3... molecules) to play a significant role as their concentration is still low. Together with the fact that the chance of emitting dimers, trimers, etc. is also low, this means that the kinetics of nucleation may be governed mainly by the attachment and detachment of monomers as a first approximation [1]. Mathematically, this is given by the restriction fnm(t) = 0,∀n, m :|n−m|>1,(3.24) Now, by imposing the above restriction into Eq.(3.18), we obtain the following equation dCn(t) dt =fn−1,n(t)Cn−1(t)−(fn,n−1(t) + fn,n+1(t))Cn(t) + fn+1,n(t)Cn+1(t),(3.25) which basically means that the concentration of n-mers depends exclusively on the attachment and detachment of monomers. Or, in other words, it depends on the next-neighbour 15 interactions. Introducing the notation k+ n≡fn,n+1 and k− n≡fn,n−1[2], and assuming that the transition frequencies are time independent, we arrive to the following expression dCn(t) dt =k+ n−1Cn−1(t)−(k− n+k+ n)Cn(t) + k− n+1Cn+1(t),(3.26) which is the core of the well known Becker-D¨oring-Tunitskii Model (BDT Model) [1, 2]. With the above equation, the flux of clusters defined in the previous subsection will be given by jn(t) = k+ nCn(t)−k− n+1Cn+1(t).(3.27) Once Eq.(3.26) is obtained, we will focus on developing a similar equation for the case of treating the cluster size as a continuous variable. To do so, we will perform truncated Taylor expansions of the terms k+ n−1Cn−1(t) and k− n+1Cn+1(t) around the n-size k+ n−1Cn−1(t) = k+ nCn(t)−∂(k+ nCn(t)) ∂n +1 22∂2(k+ nCn(t)) ∂n2+O(3),(3.28) k− n+1Cn+1(t) = k− nCn(t) + ∂(k− nCn(t)) ∂n +1 22∂2(k− nCn(t)) ∂n2+O(3),(3.29) where is a formal ordering parameter. Inserting these expressions into Eq.(3.26), setting = 1 and dropping the 3and higher contributions of both expansions we arrive to the equation ∂C(n, t) ∂t =−∂ ∂n D(1)(n, t)C(n, t)−1 2 ∂ ∂n D(2)(n, t)C(n, t),(3.30) with, D(1)(n, t) = k+(n, t)−k−(n, t), D(2)(n, t) = k+(n, t) + k−(n, t). Equation (3.30) is known as the Tunitskii Equation [2], and it has the similar form as the one for a general Generation-Recombination process (GR) 5[2]. We may identify D(1)(n, t) as the drift current of clusters along the size axis and D(2)(n, t) as the diffusion current or white noise associated to the growth velocity and randomness. It is interesting to see where this nomenclature comes from; On one hand, if we neglect the second term from Eq.(3.30), a continuity-like equation is derived, for which D(1)(n, t) represents the drift current, while on the other hand, if we neglect the first term, then the equation takes the form of a diffusionlike equation. However, the latter expression would only apply if C(n, t) was a probability function. Thus, we define then the probability of observing a n-mer as follows P(n, t) = n(n, t) Pn0n(n0, t)=n(n, t) Nc(t),(3.31) 5A common example for this kind of processes are the creation and destruction of electron-hole pairs between the valence and conduction bands within a semiconductor material. 16 where n(n, t) is the number of n-mers and Nc(t) the total number of clusters. Thus, C(n, t) = n(n,t) V=Ctot(t)P(n, t), with Ctot(t) = Nc(t) V. If the total number of molecules is constant during the transition, that is, Ncand thus Ctot are constant, then Eq.(3.30) turns into ∂P(n, t) ∂t =−∂ ∂n D(1)(n, t)P(n, t)−1 2 ∂ ∂n D(2)(n, t)P(n, t).(3.32) This equation has the form of a Fokker-Planck Equation (FPE). The cluster size nis then considered a stochastic markovian process, whose first and second Markov moments are D(1)(n, t) and D(2)(n, t). Thus, the differential equation that nmust satisfy will have the form of a Langevin Equation [3] n(t+dt) = n(t) + D(1)(n, t)dt +N(0,1)qD(2)(n, t)dt, (3.33) where N(0,1) corresponds to a random number from the normal distribution. Although we have employed several approximations, Eq.(3.32) reflects the stochastic nature of of the concentration of clusters, and thus the phase transition itself. For small clusters, the derived Fokker-Planck equation does not describe the same phenomenon as the BDT equation, Eq.(3.26), because higher order Markov moments become significant. In that case, by taking into account the whole Taylor expansions from Eq.(3.28) and Eq.(3.29), we arrive to the following expansion ∂P(n, t) ∂t = ∞ X j=1 (−1)j j! ∂j ∂njD(j)(n, t)P(n, t),(3.34) with the Markov moments D(j)(n, t) = jk+(n, t)+(−1)jk−(n, t).(3.35) This equation constitutes a Kramers-Moyal Equation (KME) [3], which is a generalisation of the master equation for continuous variables. Despite being formally exact, the KME contains an infinite number of Markov moments, for which its numerical solution becomes even more difficult, and then truncation methods are required [3]. It is also interesting to outline that we can recover the FPE by considering from the KME just the first two moments of the expansion. 3.2.2 Equilibrium Approach The KME and FPE allow us to compute the time evolution of the probability of finding a n-mer (and thus, concentration of n-mers) at any time, with the assumption of large clusters. Thus, we can use the tools of mathematical analysis and continuous calculus. However, in order to obtain P(n, t), the statistical moments associated to the Markov process and thus 17 the attachment and detachment rates are required. On the one hand, the attachment rate will depend mainly on the mechanism governing the transport of particles within the old phase, that is, a mass transport problem [2], while on the other hand, the detachment rate depends on the physical properties of the cluster, which remain unknown. Zeldovich proposed an idea in order to establish a relation between the monomer attachment and detachment frequencies [1, 2]. This relation comes from demanding that the flux of clusters defined by Eq.(3.27) to be zero under equilibrium conditions, that is, when the system is undersaturated. Mathematically, this demand is given by the condition jeq n(t) = k+ nCeq n(t)−k− n+1Ceq n+1(t) = 0,(3.36) where the superindex eq denotes the equilibrium conditions. This hypothesis is known as detailed balance or constrained equilibrium condition [2]. At quasi-equilibrium conditions, we can also consider the concentration of clusters being governed by a time-dependent Boltzmann distribution [1, 2], namely Ceq n(t)∝exp(−β∆Gn(t)),(3.37) where ∆Gn(t) is the Gibbs free energy of the system containing an n-mer. For the sake of simplicity, we will consider that the temperature will remain constant throughout the process, so that βwill be then time-independent. This assumption is known as the isothermal condition [1, 7]. By substituting this on Eq.(3.36) we obtain the following relation k− n+1(t) = k+ nexp [β(∆Gn(t)−∆Gn+1(t))] ,(3.38) which is the core of the Zeldovich approach. However, as we already mentioned, the detailed balance only holds for equilibrium conditions, so applying it for a nonequilibrium situation such as nucleation itself constitutes a huge assumption that we will discuss on further sections. Now, by expanding Eq.(3.38) in its continuous form, we obtain k−(n, t) = k+(n−1, t) exp [β(∆G(n, t)−∆G(n−1, t))] ≈k+(n, t)−∂k+(n, t) ∂n 1 + β ∂G(n, t) ∂n  =k+(n, t)−∂k+(n, t) ∂n +βk+(n, t)∂G(n, t) ∂n , (3.39) where we have just considered the first-order terms of each expansion. By substituting the above expansion into the FPE equation provided by Eq.(3.32), we obtain the following expression ∂P(n, t) ∂t =∂ ∂n k+(n, t)β∂G(n, t) ∂n P(n, t) + k+(n, t)∂P(n, t) ∂n ,(3.40) 18 which is known as the Zeldovich Equation (ZE) for the probability distribution function [2]. It will allow us to compute analytical estimations for both the cluster size distribution and the nucleation rate. We must discuss the following aspects regarding the ZE equation: •Cluster Size Limitations: In order to derive Eq.(3.40) we have employed several assumptions. On one hand, we have neglected the ∂k+(n,t) ∂n contribution by considering that the attachment rate changes very little for large clusters, while on the other hand, we have just taken into consideration the first term of the expansion of the exponential in Eq.(3.39), which is only justified for ∂G(n,t) ∂n 1. Namely, we are limiting ourselves within the vicinity of the critical cluster, for which the Gibbs free energy reaches its maximum. •Equilibrium Solution: We can impose the equilibrium situation by demanding a zero probability flux in the ZE [2, 7] βdG(n) dn Peq(n) + dPeq(n) dn = 0.(3.41) By solving the above ordinary differential equation, we obtain Peq(n) = Aexp(−β∆G(n)) →Ceq(n)∝exp(−β∆G(n)),(3.42) where the constant Ais fixed by the normalization of the probability distribution function Peq(n). This last expression is useful in order to check the canonical form of the equilibrium distribution [2], as we firstly assumed with Eq.(3.37). •Stationary Solution: The physical situation underlying this solution is that the system reaches a steady state where the cluster production is constant. If a cluster reaches the supercritical size, it will grow indefinitely, so it will no longer be considered a stochastic process. In order to impose a stationary flux Js, we can remove those clusters greater than the supercritical size (n+> n∗) and reintroduce them in the form of monomers [2]. Mathematically, the restricted stationary flux is given by Js=−k+(n)β∂G(n) ∂n Ps(n) + k+(n)∂Ps(n) ∂n ;Ps(n+)=0.(3.43) Solving the ordinary differential equation given by the above equation and applying the given initial condition, we obtain the stationary probability distribution function [2] Ps(n) = Jsexp(−β∆G(n)) Zn+ n dx k+(x) exp(−β∆G(x)),(3.44) which will play a key role in the calculation of the nucleation rate. 19 3.2.3 Nucleation Rate in CNT At this point, we have developed the theoretical background and all the mathematical tools that will allow us to compute an analytical value for the most important magnitude underlying the CNT; the nucleation rate, which measures the production of new clusters per unit volume. Its importance lies within the fact that the nucleation rate can be measured experimentally, thus establishing a first connection between the theory and the real world. It also allows us to check the validity of all the presented ideas by comparing them with the experimental results. By taking into account that only supercritical clusters will take part in the nucleation process, the nucleation rate is defined by [2] J(t) = dCn>n∗(t) dt ,(3.45) where Cn>n∗(t) is the concentration of supercritical clusters, that is, those clusters whose sizes are greater than the critical one. Within the continuous approach, Cn>n∗(t) is given by Cn>n∗(t) = Z∞ n∗ C(n, t)dn =Z∞ n∗ Nc(t) VP(n, t)dn, (3.46) with Nc(t) = Z∞ 1 n(n, t)dn, (3.47) being the total number of clusters [1, 2]. Now, by inserting Eq.(3.46) into Eq.(3.45), we obtain J(t) = 1 V dNc(t) dt Z∞ n∗ P(n, t)dn +Nc(t) VZ∞ n∗ ∂P(n, t) ∂t dn. (3.48) Using the Zeldovich Equation in the second term and imposing the steady state condition, we arrive to the following expression J=−Nc Vk+(n)β∂G(n) ∂n Ps(n) + k+(n)∂Ps(n) ∂n =Nc VJs,(3.49) which basically means that the nucleation rate depends mainly on the artificially imposed stationary flux Jsfrom the previous subsection. In order to go further, we need to develop an analytical expression for Js. Zeldovich proposed a boundary condition in order to solve this problem [2]; monomers are indeed particles from the old phase, so their steady concentration must be equal to the initial average density Cs(n= 1) = ρav.(3.50) 20 Following this idea, the steady probability of finding monomers will be given by Ps(1) = Cs(1) Ctot =ρavV Nc ,(3.51) and thus, by substituting this into Eq.(3.43), we arrive to Js=ρavV Nce−β∆G(1) Zn+ 1 dx k+(x) exp(−β∆G(x))−1 .(3.52) The first exponential term comes from the fact that the system does not need to spend energy in forming monomers, as they already belong to the old phase. Now we will approximate the above integral by means of the Lagrange saddle-point method [8]. Knowing that the exponential term exp(β∆G(n)) must show a sharp maximum at the critical size n∗, we can use the following first order asymptotic expansion [1, 2] ZIx exp(β∆G(x)) k+(x)dx ≈exp(β∆G(n∗)) k+(n∗)−β 2π ∂2∆G(n) ∂n2−1 2 n=n∗ +O(β−1).(3.53) Finally, by applying these expressions on Eq.(3.49), we arrive to an analytical expression for the nucleation rate within the CNT context JCNT ≈zdk+(n∗)ρav exp −W∗ kBT,(3.54) where W∗is the minimum work required for a critical cluster to form, and zd=−β 2π ∂2∆G(n) ∂n2−1 2 n=n∗ ,(3.55) is the Zeldovich factor [2], which can be also expressed by means of the simplified formula zd=βW∗ 3π(n∗)21 2 .(3.56) However, in our analysis we have not considered that the condition given by Eq.(3.50) implies that monomers already belonged to the old phase before the transition, so they do not contribute to the energy barrier W∗. In order to take this into account, we must redefine the energy barrier as W∗ SCCT =W∗−W(n= 1).(3.57) The sub-index SCCT comes from the Self Consistent Classical Theory, which provides an expression for the nucleation rate given by JSCCT ≈zdk+(n∗)ρav exp −W∗ SCCT kBT.(3.58) 21 Let us briefly discuss the physical meaning underlying both Eq.(3.54) and Eq.(3.58). First, we must notice that this expression contains the same exponential term as the one we initially derived in previous sections from the thermodynamic approach. It is related to the probability for a critical cluster to form. Also, the initial average density ρav is related to the physical properties of the system in its old phase. In addition, we have the Zeldovich factor, which is directly related to the nucleation barrier, and the attachment constant k+(n), which provides us information about the mass transport phenomenon that is taking place in the system [1]. However, the dominant term of the expression is the exponential one, which tells us that, in the end, the nucleation rate is primarily governed by the nucleation barrier. This fact is corroborated by the theorem developed by Kashchiev [1] within the context of the CNT, known as the Nucleation Theorem, which relates the dependency of the nucleation rate with the supersaturation by means of the nucleation barrier and the critical nucleus. Mathematically, it is expressed by ∂log J ∂log S≈ −∂βW∗ ∂log S=n∗,(3.59) where Jrelates to the nucleation rate (in both the CNT and SCCT) and Sis the supersaturation. Its importance lies in the fact that, given that the nucleation rate, the amount that allows us to make a connection between theory and experiments, it depends mostly on the physics behind the nucleation barrier, so a good step to begin in order to improve the classical theory would be to consider a new paradigm of nucleation from which corrections can be made to this barrier, and with this, the density functional theory would come into play. In conclusion, the nucleation rate depends mainly on the nucleation barrier, and secondarily on the properties of the system before the transition occurs and the phenomenon through which particles within the system can diffuse and collide. 3.3 Homogeneous Nucleation of Water Once we have developed the theoretical framework that allows a classical description of the nucleation process, we will use its main results in order to investigate the case of timeindependent homogeneous nucleation of water droplets in the vapor phase. We have chosen water due to its importance in atmospheric processes, which play a key role in the development of life on Earth. In this section we will study the equilibrium probability functions from both the thermodynamic and stochastic approaches, and we will compare them by means of the nucleus critical size. Also, we will obtain the nucleation rates of water droplets for different supersaturations 22 Despite being a simple model, it does not represent a real physical situation. In order to improve the model, we shall introduce what are known as n-parameter theories [2], for which clusters are completely defined by a set of ndifferent parameters (the radius, width of surface, etc), and thus, the density profile adopts a more realistic form, such as in Figure 3(a). Another more sophisticated possibility that we will discuss in further sections is the Density Functional Theory (DFT), which will allow us to obtain the density profile by means of calculus of variations [1]. The only difficulty that arises from those approaches is that we would need to reformulate the entire kinetic theory of nucleation, because it is based mainly on the assumption of well defined clusters that provides the capillary model. •Large Clusters Assumption: The thermodynamic approach introduced in section 3.1 in order to develop an expression for the nucleation barrier are only valid for a large number of molecules compounding the cluster, that is, in the thermodynamic limit [7]. Also, in order to derive the Fokker – Planck and Krammers – Moyal equations, we considered that the cluster size was large enough so it may be treated as a continuous variable. Indeed, together with the restriction imposed by the equilibrium approach, for which the range of validity of the nucleation rate lies within the vicinity of the critical size, it is though true to confirm that the description and contribution of small clusters to the nucleation process is outside the scope of the CNT. A self-consistent theory of nucleation must be able to describe both kinetics and energy balance of clusters within the old phase no matter their size [2]. •Equilibrium Approach: With the Zeldovich approach we were able to solve the problem of the detachment rate of clusters by imposing equilibrium conditions on the undersaturated system, resulting in an expression that relates this detachment rate with the attachment rate, which is a more suitable magnitude to compute. However, this argument leads to unphysical situations, because we are imposing that the new phase is in equilibrium with the old metastable phase during the transition, where actually the nucleation itself consists in a non-equilibrium process [1]. Moreover, the derived equilibrium probability distribution functions are only normalizable when the system is undersaturated [2], which, again, it is a nonphysical situation in our case. •Lack of Covariance: In previous sections we used two different parameters for the description of the clusters size: the number nof molecules and the radius Rof the associated spherical cluster. Although both descriptions are unequivocal, the interchange is not symmetric when we are dealing with the equilibrium probability distribution functions P(n) and P(R) respectively [2]. 29 On one hand, with the Zeldovich approach, under equilibrium conditions, we derived a distribution function proportional to the Boltzmann factor P(n)∝exp(−β∆G(n)).(3.69) While on the other hand, within the framework of the thermodynamic approach, we also derived an expression for the minimum work required for a cluster to form, which is directly related to the amount of Gibbs free energy required for the process to occur. The problem arises when we want to switch between distribution functions, because they are actually related by the transformation relation P(n)dn =P(R)dR, (3.70) which means basically that the probability of finding clusters with a certain size must not depend on the chosen variable for its description. Using the above equations, we can estimate the probability distribution function for the spherical radius P(R)∝exp −β∆G(R) + kBTln dn(R) dR ,(3.71) which is similar to the ones used in previous sections but for the Jacobian term. For large clusters, this contribution can be neglected in comparison with the volume and surface terms, but again, we are limiting ourselves to a certain range of validity for the cluster size. 4 Density Functional Theory In the previous section we developed the Classical Nucleation Theory, which is fundamentally based on the definition of a cluster as a portion of the new phase formed within the old one during the phase transition. This cluster will grow or sink through the attachment and detachment of monomers, depending mainly on its initial size. With these ideas, all the theoretical background underlying the theory was developed. However, as it was already discussed, some physical issues emerge from the main assumptions employed in the development of our theory, one of the most important ones being the definition of a cluster. Within the context of the Capillary Model the answer is simple; all the molecules belonging to the region lower than the equimolar dividing surface form a cluster, while the rest of them conform the environment, that is, the old phase, as depicted in Figure 3b. However, we already pointed out the abruptness and simplicity of this model. A more realistic density profile for the cluster corresponds to the one shown in Figure 3a, where the change in density 30 is continuous. Thus, the density will depend on the spatial coordinates, that is, ρ=ρ(r). A completely new theoretical approach is then necessary in order to obtain better estimations for the nucleation barrier. Limiting our considerations to one-component homogeneous nucleation under isothermal and isobaric conditions, we introduce the Density Functional Theory (DFT) of nucleation, based entirely on the density functional approach [1]. 4.1 Density Functional and Variational Approach As it can be seen in Figure 3a, instead of a dividing surface between the molecules of the new and old phases, we have a continuous change of the molecular density through a transition region known as the surface layer [1]. Let ρ(r) be the molecular density at point r= (x, y, z) in cartesian coordinates. This density is related to the average density ρav through the expression [2] ρav =1 VZV ρ(r)dr.(4.1) Due to the fact that ρ(r) is non-homogeneous, other quantities related to it will present dependence with spatial coordinates too. Such is the case of the Helmholtz free energy per molecule f, which becomes a function of r,ρand its derivatives. That is the main reason why thermodynamics, which operates with homogeneous quantities, is not adequate for determining the nucleation work W∗. Hence, we must reformulate the Gibbs free energy for both initial and final states of the system (Figure 2) as a function of the molecular density function. We will define the total free energy as a functional of the density given by G[ρ] = ZV g(ρ(r), T)dr=ZV (f(ρ(r)) −µρ(r))dr,(4.2) where g(ρ(r)) constitutes the Gibbs free energy per unit volume at constant temperature and µis the chemical potential. According to the definition presented in previous sections, the minimum work for a density fluctuation within the old phase will be given by W[ρ] = ∆G[ρ] = Gf[ρ]−Gi[ρ] = ZV ((f(ρ(r)) −µρ(r)) + p)dr,(4.3) where we have taken into account that Gi[ρ] = −RVpdr, approaching Eq.(3.3) in the thermodynamic limit. The resulting expression is valid for any arbitrary density profile. Similar to how we proceeded with the thermodynamic approach, we will focus our attention into the critical fluctuation (so called nucleus [2]), that is, the one required for triggering the phase 31 transition. It can be defined as a saddle point in functional space. To obtain the density profile ρ∗(r) characterizing the critical fluctuation, we must solve the following variational problem (δW[ρ])ρ=ρ∗≡δZV ((f(ρ(r)) −µρ(r)) + p)drρ=ρ∗ = 0,(4.4) whose solution will allow us to determine the nucleation barrier W∗≡W[ρ∗(r)].(4.5) The procedure presented above allowed us to obtain the variational equations required for the nucleation barrier without the need of introducing neither a dividing surface nor a capillary model in order to distinguish between the nucleus (cluster) and the old phase. In this case, the first stage of nucleation is treated as a density fluctuation of the old phase. This is in contrast with our previous analysis, where the cluster formed within the old phase already presented the same properties as the new phase and it was treated as a macroscopic independent system. The only problem regarding Eq.(4.4) is that we need the exact form of f(ρ(r)) in order to obtain a solution for the density profile. However, even without having this information, we can derive a partial differential equation from this equation by considering the Helmholtz free energy as an exclusive function of ρand ∇ρ. In that case, the condition of extremal W[ρ] only applies if the Euler-Lagrange equation is satisfied [1, 2] ∂ ∂ρ(f(ρ, ∇ρ)−µρ)− 3 X i=1 ∂2 ∂xi∂ρxi (f(ρ, ∇ρ)−µρ)!ρ=ρ∗ = 0.(4.6) Here xiare the cartesian coordinates, components of r, and ρxi= (∇ρ)xi. We can transform the obtained equation into the following one ∂ ∂ρf(ρ, ∇ρ)− 3 X i=1 ∂2 ∂xi∂ρxi f(ρ, ∇ρ)!ρ=ρ∗ =µ, (4.7) which can be interpreted as the chemical equilibrium between the fluctuation and the old phase [2]. The equation corresponds to a second order differential equation. Its physically acceptable solution must fulfill certain boundary conditions at the surface of the volume V of the system. In the case of spherical symmetry, it is interesting to outline the following boundary conditions commonly employed lim r→∞ ρ∗(r) = ρav,dρ∗(r) dr r=0 = 0,(4.8) which apply for the two density profiles showed in Figure 3. Once we have the exact solution (or a good approximation) for the Helmholtz free energy, the determination of the critical fluctuation ρ∗(r) and the nucleation barrier W∗are guaranteed. 32 4.2 Approximations for the Free Energy We have seen previously that the equations describing the critical fluctuation depend mainly on the modelling of the Helmholtz free energy. This modelling must take into account the non-local effects; the interaction between molecules at rand r’ respectively. One way of doing that is considering fnot only as a function of the unary density (i.e ρ(1) ≡ρ), but also the binary ρ(2), ternary ρ(3), etc [1]6. However, this procedure implies several mathematical difficulties due to its complexity, so it is necessary to apply certain approximations. In this work we will focus our attention on two approximations; the Gradient Approximation (GA) and the Hard-Sphere Model (HSM), which, despite their simplicity, they will allow us to recover results from previous sections regarding the CNT and the VDW fluid. 4.2.1 Gradient Approximation This first approximation consists in considering the free energy as an exclusive function of ρ and its first derivatives. Its general form is given by f(r) = fu[ρ(r)] + K[ρ(r)](∇ρ)2, K[ρ(r)] >0,(4.9) which corresponds to a truncated expansion of f(r) in terms of the density gradient ∇ρ. In the above equation, fuwould correspond to the zeroth order term of the expansion, which is the global Helmholtz free energy per molecule, namely, the one the system would have if local effects were not involved, while the second term would correspond to the second order term of the expansion, accounting on the interactions between molecules at rand the rest of the environment, making f(r) a non local function of r[2]. The dependence of the functions fuand Kon ρ(r) can be determined by means of the regular solution theory of mixtures, as pointed out by Kashchiev [1]. By substituting Eq.(4.9) into Eq.(4.7), we are led to the Cahn-Hilliard Equation (CHE) ∂ ∂ρ(fu(ρ)) −(∇ρ)2∂ ∂ρ(K(ρ)) −2K(ρ)(∇2ρ)ρ=ρ∗ =µ. (4.10) This last equation can be simplified for the case of a spherical nucleus, as the critical fluctuation will only depend on the radial distance r,ρ∗(r) = ρ∗(r). Considering that in spherical coordinates the following expressions apply (∇ρ)2=dρ dr2 ,∇2ρ=d2ρ dr2+2 r dρ dr,(4.11) 6It is defined the n-density ρ(n)(r(n)) as the density that takes into account the effect of an n-wise potential over the system. In case that n= 1, we recover the density for an ideal solution [6] 33 the CHE adopts the following form d dρ(fu(ρ)) −2K(ρ)d2ρ dr2−4K(ρ) r dρ dr −dρ dr2d dρ(K(ρ))!ρ=ρ∗ =µ, (4.12) By applying the boundary conditions presented in Eq.(4.8) to the above expression, the critical fluctuation can be computed numerically, and thus, the nucleation barrier can also be determined by solving Eq.(4.3), considering that dr= 4πr2dr due to the spherical symmetry. It is interesting to outline three results coming from the numerical simulations carried out by Cahn and Hilliard [1] using this model for low supersaturations: •The density at the centre of the nucleus (ρ∗(0)) approaches the one characterising the new phase (ρnew). •The specific surface energy associated to the surface layer of the nucleus reaches the planar surface one. •The appropriately defined mean radius approaches the one defined by the equimolar dividing surface. Summarising, these aspects confirm the validity of the model for low supersaturations, as it leads to the same results as those predicted within the context of the CNT. However, the behaviour of both theories differs significantly for high supersaturations. 4.2.2 Hard-Sphere Model This model was first introduced in order to avoid the GA, which is only valid for smooth variations of the density profile [1]. The HSM is based mainly on the proposal of an expression for the free energy analogous to the one derived for the VDW fluid within the thermodynamic context fV DW =f0+kBTln b0ρu 1−b0ρu−a0ρu,(4.13) where f0and ρuare reference energy and density respectively, and a’ and b’ are the same material constants presented in Eq.(2.1). b’ accounts for the harsh short-range repulsion between molecules when they are closer than a radial distance d0, and a’ considers the weak long-range attraction between them, and it is generally given by the expression a0=−2πZ∞ d0 r2u(r)dr, (4.14) 34 where u(r) is the pairwise interaction potential [1]. For our case, a non-homogeneous density profile ρ(r), the VDW-like expression for the Helmholtz free energy will be given by f(r) = fHS[ρ(r)] + 1 2ZV0 ρ(r’)u(|r−r’|)dr’,(4.15) where the first term corresponds to the free energy associated to a hard-sphere potential, while the second one accounts for the attraction between molecules at the point rwith the rest of the molecules of the system. By substituting the above expression in Eq.(4.7), we obtain the following equation µHS[ρ(r)] + ZV0 ρ(r’)u(|r−r’|)dr’ρ=ρ∗ =µ, (4.16) where µHS[ρ(r)] = ∂ ∂ρ (fHS(ρ)). Solving this equation with the previous boundary conditions, the critical fluctuation and the nucleation barrier can be computed. But in order to compute such quantities, we need a concrete form of the mean field pairwise potential u(|r−r’|), like the Yukawa or Lennard-Jones potentials [1]. In comparison with Eq.(4.12), Eq.(4.16) is an integral equation that does not include ∇ρ, but it allows us to obtain nearly-step solutions for the density profile, so that comparisons with the capillary model already presented with the CNT may be carried out. 4.3 N-Parameter Theories In previous sections we have obtained differential equations that allow us to compute the critical density profile that minimizes the nucleation barrier, depending on the physical situation we are dealing with, for which we must specify different functions such as the Helmholtz free energy or the chemical potential. However, despite its usefulness, in most cases only numerical results will be obtained, while it is more suitable for physicists to work with analytical functions. These functions, provided by theoretical models, will depend on a set of parameters that will be adjustable in order to fit the experimental measurements. We will assume that the density profile has already a certain analytical form, which depends on a set of nparameters X=X1, X2, ...Xnsuch as follows, ρ=ρ(r,X) = ρ(r, X1, X2, ..., Xn).(4.17) In this case, due to the fact that we already set a model density function, our goal now consists in determining the critical parameters X’ =X0 1, X0 2, ...X0 nthat minimize the nucleation barrier, and then it can be proved that the variational problem presented in Eq.(4.4) transforms into (∇XW(X))X=X’ = 0,(4.18) 35 where we have used the notation ∇X=Piui∂X[2]. This method is widely used because of the reduction of the computational cost needed to perform numerical calculations. The analytical models for nucleation based on this methodology are known as n-parameter theories. We have already studied an example of this kind of model; the capillary model constitutes itself a clear example of an one-parameter theory, for it assumes a density profile that only depends on the radius of the spherical cluster. Thus, the radius Rof the cluster constitutes the order parameter of the model7. With this, the nucleation barrier was obtained with Eq.(3.10), which is analogous to Eq.(4.18). Furthermore, the entire CNT constitutes an one-parameter theory, due to the fact that we assumed that the clusters are completely determined by one parameter; the nucleus size (or the radius for spherical nuclei). Another example of these models is given by the piecewise density profile ρ(r) =          ρnew r≤R−w ρnew + (ρnew −ρav)r−(R−w) wR−w≤r≤R ρav r≥R (4.19) which takes into account a gradual linear variation of the density through the surface of the spherical cluster, which has a width w. In this case, both the spherical radius Rand the surface width wwould be order parameters, and thus we would encounter a two-parameter theory. 5 Conclusion Finally, we have arrived to the end of our journey. In this last section we will summarize the aspects that we have developed throughout this work, pointing out the main ideas and results. We first started with the basis of first-order phase transitions. We used the Van Der Waals fluid as our key example in order to understand vapor-liquid transitions, and with this, the equilibrium conditions needed for the coexistence between phases, and the concepts of metastability, spinodal and binodal curves and supersaturation were introduced. Gathering all of these with a brief introduction to thermal fluctuations, we arrived to the definition of 7Despite that the density profile given by Eq.(3.8) also depends on the parameters ρnew and ρav, these parameters do not count as order parameters, for they are initial conditions that can be obtained experimentally, and are, in principle, independent from the nucleation process. 36 the nucleation process itself. From here, we focused our analysis into the Classical Nucleation Theory. Starting with Gibbs ideas, we defined a cluster as a thermodynamic system surrounded by a greater one, and with the help of the capillary model and the assumption of spherical clusters, we derived an expression for the nucleation barrier, and thus also for the probability for a density fluctuation to occur. Then we studied the kinetic model provided by the CNT. Starting with the definition of the cluster and the transition frequencies, we arrived to the Master Equation of Nucleation. From here, with the aid of the BDT Model and Zeldovich’s ideas, we derived Fokker-PLanck, Languevin and Kramers-Moyal equations that allow a stochastical description of the nucleation process. Following this, considering the steady state, we arrived to an analytical expression for the nucleation rate, a quantity that can be measured experimentally, and thus, it links the theory and reality. In order to apply the previous results, we analysed the homogeneous nucleation of water droplets in condensed vapours. We first solved the Langevin Equation with the aid of the Euler-Maruyama approximation, leading to the equilibrium histograms. Then, we compared the theoretical nucleation rates with experimental ones measured in diffusion cloud chambers, leading to several inconsistencies of the theory. All of this analysis was useful for us to link with a last section of the CNT where its main deficiencies were commented. Following the CNT, the Density Functional Theory was introduced as an alternative paradigm for the determination of the nucleation barrier and the definition of clusters themselves. Variational principles were used for deriving the nucleation barrier, and two approximations for the free energy as a functional of the density were introduced. These approximations also allowed us to recover previous aspects provided by the CNT and the VDW fluid. Finally, the concept of n-parameter theory was introduced as an analytical alternative to the numerical calculation, which allowed to present theoretical models depending on the process under study. In conclusion, the field of nucleation is still an open field to study, where there are many unanswered questions yet, and which still holds many mysteries of matter and the world itself. 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[8] Nicolaas Govert de Brujin, Asymptotic Methods in Analysis, North Holland, 1998. [9] Jean-Pierre Hansen, I.R. McDonald, Theory of simple liquids, Elsevier Academic Press, 2006. [10] A. A. Lopatkin, Van der Waals Equation, The Great Soviet Encyclopedia, 3rd Edition, Moscow, (1970-1979). [11] H.P. Lovecraft, The Call of Cthulhu, Weird Tales, New York, 1928. 38