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Fluctuation effects in metastable states near first order phase transitions

Fluctuation effects at first order phase transitions driven by changes of other-than-temperature factors like pressure, concentration, or external fields are investigated by perturbation theory. The results for the fluctuation contributions to the order parameter, the internal energy and the free energy at pre-transitional states near spinodal points of first order phase transitions are presented to the first-non-vanishing order of the expansion parameters of the theory.


Introduction
From the stand point of the theory of phase transitions the above mentioned examples belong to the category of first-order transitions driven away from the point of equilibrium. One of the best approaches to consider these processes is the Ginzburg-Landau (GL) theory, which is based on the concept of order parameter as the main characteristic of the phase transition and the phases [1][2][3][4]. The generalized GL free energy (fluctuation Hamiltonian) is a functional of the order parameter field and can be suitably chosen to incorporate the most essential thermodynamic features of the system including the relevant fluctuation effects.
Traditionally, fluctuations around stable thermodynamic states have been studied in the framework of GL Hamiltonians of the type 2-4 where the numbers indicate the powers of the order parameter in the truncated power-series expansion [5,6]. In the fluctuation theory of phase transitions [2,3] the quadratic term is often called "Gaussian term" whereas any non-quadratic term is called "interaction term", e.g., the quartic term in 2-4 Hamiltonians. The quadratic term describes free (non-interacting) fluctuation modes whereas the terms of higher order in powers of the order parameter field describe the relevant fluctuation interactions in the system [2,3].
The usual 2-4 theories describe standard second order phase transitions where metastable states do not appear. To study metastable states and related problems of materials physics, e.g. phase nucleation, growth, and coarsening, many researchers use the GL approach to Hamiltonians of type 2-3-4, where an additional term of third order in the order parameter is included into consideration [3,6]. Hamiltonians of this type describe first order phase transitions between stable phases. Under certain thermodynamic conditions the same phases may appear by metastable states or to coexist in equilibrium. Owing to the availability of the cubic interaction term, the fluctuation properties of the 2-3-4 Hamiltonians are quite different from those of the more common 2-4 Hamiltonians. Another essential difference with common cases arises in systems when the phase transition is not driven by the temperature but rather by the variations of another thermodynamic parameter. In such cases, the (Landau) parameters of the 2-3-4 theory have unusual temperature dependence and, hence, some thermodynamic properties, including fluctuation effects will differ from those corresponding to more common cases.
In order to investigate such cases, in this paper we introduce an essentially new dependence of the 2-3-4 Hamiltonian on the temperature and the thermodynamic parameter whose variations transform the thermodynamic states from one another and, hence, are responsible for the phase transition of first order. This model could be used to describe a variety of real systems undergoing a certain type of first order phase transitions and metastable phases which occur at fixed temperature and are driven by other-than-temperature factors (pressure, concentration, etc.). Our attention is focused on the fluctuation properties of the metastable states. We calculate the free energy, the internal energy, and the order parameter outside the close vicinity of the spinodal point where the fluctuations are anomalously strong and the perturbation theory breaks down [3].
The paper is organized as follows. In Sec.  The general GL Hamiltonian of such systems with short-range intermolecular interactions can be written in the general form [1][2][3][4] , (1) where the integration over the spatial vector x is performed in the volume V of a three dimensional system, the gradient term with the nabla operator ) and the stiffness parameter  represents the Within the MF approximation [2,3], the equilibrium values e  of the total uniform order parameter  are obtained as solutions of the equation H and the MF free energy is given by Besides, we remind that the extrema of the GL functional could be of three types: (1) global minimum, namely, the stable phase, (2) relative minima (metastable phase), and (3) maxima or inflection points, which correspond to unstable phases. These stability properties of the phases may change with variations of the theory parameters, in our case, the parameter  . For example, suitable variations of the tuning parameter  may transform the stability feature of a given phase from stability to metastability, or, even to instability, and vice versa. Phases which occur in real situations are the stable phases (under usual thermodynamic conditions) and metastable phases (under some particular conditions). The unstable phases are practically impossible to occur and therefore uninteresting.
To obtain concrete results we should shape the density of a particular form. A broad set of first order phase transitions with scalar order parameter field can be described by the 2-3-4 Hamiltonian Here H  is the Hamiltonian density of the disordered -phase (  =0), W is an energy scale parameter introduced for convenience. The equilibrium phases   =0 and ordered (β-phase)   =1 are obtained as solutions, and, therefore, describes a stable phase.
can easily be obtained: we find the MF free energies of the phases and  :

General scheme of calculations
For our purposes we present the Hamiltonian (1)-(2) in the form , where is the α-phase free energy, is the quadratic ("Gaussian") part describing free fluctuations, and -the interaction part, is given by two terms: (4b) and . (4c) The statistical average of any physical quantity is defined by (5) where , is Boltzmann's constant, T is the temperature, and is the partition function. In Eqs. (5) and (6), denotes the functional integration over all possible configurations of the field η(x). For short, in the reminder of this paper we will often denote various functionals, for example, and , by omitting the argument x of the field , and , or, even by omitting the argument at all: and .
The functional integrals in Eqs. (5) and (6) GA is unreliable when the fluctuation interactions are relatively strong. This usually happens near critical points (continuous phase transitions) and spinodal points of first order phase transitions [3]. In our case, this occurs in a close vicinity of the spinodal point ( 1). In order to perform a more thorough study of the fluctuation effects we should take into account the fluctuation interactions by developing a perturbation expansion for the relevant physical quantities [2,3].
The perturbation expansion is developed on the basis of statistical ensemble of free (non-interacting) fluctuations, namely, the ensemble defined by . For example, the exponential statistical weigh in Eqs.
Furthermore, we present the total partition function of the system as a triple product Z is given by Eq. (7), and (10) is entirely interaction product.

Perturbation expansion of several physical quantities
Within this scheme we calculate the ensemble averages, the overall (volume-averaged) order parameter, the internal and free energies. The volume-averaged order parameter is defined by .
The Gaussian average of this quantity is equal to zero ( The Helmholtz free energy is given by the sum (16) where F  =E  =VH  is the free energy of phase , is the free energy in GA, and is the fluctuation-interaction contribution to the free energy up to the second order of the perturbation expansion.

Calculations in the space of wave vectors
It is more convenient to calculate the averages in the reciprocal space of the wave vectors k. So, we apply the Fourier transformations of the order parameter field, , T is the equilibrium phase transition temperature. In this case, the correlation length has a minimal size at 0  T and is called "zero-temparature correlation length", or, sometimes, coherence length: This length defines a limit of application of the theory: The GL approach cannot be applied to studies of phenomena of characteristic size lower than the length In our case this is the "coherence radius" 0 R . Within our model, the length 0 R is therefore a candidate for a minimal length previously denoted by a .
From point of view of microphysics of particles or spin lattices, the length 0 R can be identified with the radius ( int R ) of inter-particle or inter-spin interaction responsible for the phase transition. This is seen from derivations of GL functional from microscopic Hamiltonians (see, e.g., Ref. [3]). A length which is equal or lower than the radius of particle interaction int R is the mean inter-particle distance 0 a , or, that is the same, the lattice constant of crystal bodies. For the most common case of short-range inter-particle interactions ( int R 0 a ), we may think for the minimal length 0 R in our model as the radius of interaction of real particles in the system. Alternatively, if the system is divided in mesoscopic blocks (domains) of equal size a , we could interpret the length 0 R as the radius of short-range interactions between the blocks. A block may contain a single particle, several particles and even several tens or hundreds of particles and form a quasi-macroscopic system according to the standard macroscopic approach of description. The essential point is that the number of blocks should be enough large to ensure a justification of the statistical mechanics of fluctuations and an approximate but reliable application of the continuum limit ( ). We assume that the number of blocks is 3 / a V N  is large, as given by Eq. (20) and the continuum limit could always be applied. Under these stipulations it seems reasonable to adopt 0 R for the interaction radius of blocks of size a and assume, in case of short-range inter-block interaction, that (almost "nearest neighbor block interactions"). In the extreme case of a block per particle, we have to use the relation . Note, that expansion in the ratio has been considered in Ref. [7] and, more recently, in Ref. [8].
The functional integration in integrals like those given by Eqs. (7)

Perturbation theory results
To calculate the interaction corrections to the physical quantities we apply the standard methods of the perturbation theory [2,3]. Bearing in mind that the Gaussian averages of odd products of the field are equal to zero, we can write the Eq. (12) in the form , where terms of order have been neglected.
The calculation of the averages in Eq, (33) leads to the following result , where the perturbation integrals and are given by the general formula , In order to perform a correct study we have to understand the leading dependence of the perturbation terms on the parameters and δ. Apart of any calculations we could understand this with the help of the following qualitative analysis. The shape of the correlation function (22) implies that and .
The last relation indicates that dk . Then one easily obtains that the leading term in