The diagram of phase-field crystal structures: an influence of model parameters in a two-mode approximation

Effect of phase-field crystal model (PFC-model) parameters on the structure diagram is analyzed. The PFC-model is taken in a two-mode approximation and the construction of structure diagram follows from the free energy minimization and Maxwell thermodynamic rule. The diagram of structure’s coexistence for three dimensional crystal structures [Body-Centered-Cubic (BCC), Face-Centered-Cubic (FCC) and homogeneous structures] are constructed. An influence of the model parameters, including the stability parameters, are discussed. A question about the structure diagram construction using the two-mode PFC-model with the application to real materials is established.


Introduction
The phase-field crystal model (PFC-model) is suitable for simulations of the dynamics of atomic densities on the diffusion time scales. This model can be used to calculate the coexistence of structures having various crystal lattices and to model a wide spectrum of processes [1].
The amplitude's approximation for the PFC-model is based on the limitation of the periodic atomic density field by the amplitude envelope as a non-conserved order parameter. The PFCmodel [2][3][4][5] was formulated to describe continual transitions from the homogeneous to the periodic crystal state (similarly to the Landau-Brazovskii transition [6][7][8]) and between different periodic states [9] over diffusion times. The model can be derived as a reformulation of the Swift-Hohenberg equation for the thermal fluctuation fields [10] or it follows as a first approximation of the Density Functional Theory [11][12][13]. The model uses the free energy as functional of the atomic density field which is n-periodic in the solid (crystal) phase and homogeneous in the liquid state. Periodicity of the field n naturally takes into account the elastic energy and symmetry of crystals that is described as the motion equation for the conservative order parameter. In such a way, it is possible to simulate a wide class of phenomena including, e.g., epitaxial growth and ordering of nanostructures on micrometer scales [9], crystallization and high-speed regimes of front propagation [14,15], the motion of dislocations and plastic flow, the formation of a disordered amorphous state, premelting of grain boundaries, crack spreading, rearrangement of microscopic structure of interfaces, and the dynamics of colloidal systems and polymers [16] The determination of equilibrium structures and their coexistence in "average atomic densitytransition driving force" diagrams takes a special place in the phase-field model. Such diagrams determine the structures to which unstable or metastable states of a material must evolve. Diagrams of two dimensional crystalline structures for transitions from an unstable state were constructed using the atomic density functional, amplitude expansion and the Maxwell equalarea rule [3,17]. The structural diagram for transitions for two-mode functionals from the metastable state was also constructed earlier for different control parameters [1,[18][19][20][21][22]. In the present study, for the structure diagram construction, we use a numerical method for minimizing of free energy functionals. This method makes it possible to determine the parameter range for the existence of three dimensional structures in materials. Using the free energy functional in two-mode approximation (that extends our previous study [23]), the construction for the coexistence of homogeneous (liquid) phase, Body-Centered-Cubic (BCC), Face-Centered-Cubic (FCC) crystal lattices of materials is made. The structure diagram is constructed in coordinates "temperature -atomic density".

Free energy for crystalline states
To evaluate the various properties of the different crystal states it is useful to use two-mode approximation (accounting a second atomic coordination sphere) instead of the original onemode PFC approximation. Using this approximation we can construct the set of the free energy functionals for the given phases in the equilibrium state. This approach lead to the more accurate and exact calculations for complex crystal structures.
The free energy functional for the first-order phase transformations is given by [1,18,20] where the n = (ρ − ρ liquid )/ρ liquid is dimensionless atomic density, ρ is the local atomic density, ρ liquid is the reference atomic density of the homogeneous state (liquid), ∆B 0 = B 0 − B x 0 is the driving force taken as a difference of compressibility and elastic modulus of system, a and v are phenomenological parameters which can be selected for certain type of material. To describe the energy of crystal we introduce the nonlinear operator D i : where r 0 and r 1 are shifts of the first two wave vectors, responsible for the relative stability of structures, and q 0 and q 1 are the modules of first two sublattice wave vectors. This extension of the previously introduced two-mode PFC model (Wu et al. [20] with r 0 = 0) has been proposed by Asadi and Zaeem [21]. The role of the parameter r 0 will be revealed below, here we just can mention that r 0 can be used for the exact minimization of the free energy by the parameter q to find the wave number of the equilibrium state for the crystal or homogeneous (liquid) phase.
To get a compact and dimensionless form of the free energy (1), it is convenient to use the following substitutions: Hence the free energy (1) transforms to where we neglect the high order terms for the simplicity. Then the dimensionless operators D i (2) are: with As a result, the free energy (4) with D 2 from eq. (5) has three parameters: R 0 , R 1 and ε. The first two parameters control the stability of structures. The third parameter, as the driving force parameter ε, is the relative temperature ε = (T c − T )/T which determines the excess over the transition temperature T = T c and the relation between elastic properties of the system in terms of quantities ∆B 0 and B x 0 . In the next section we will use the two-mode form of the functional which implies the form of the amplitude's expansion coefficients and number of the parameters.

Crystalline structure determination and selection of parameters
Using the summation of reciprocal lattice vectors [1,19,21] we can make the amplitude expansion and get the dimensionless density profiles indicating the symmetry and properties of a given phase (homogeneous, BCC, FCC): n BCC =ñ + 4η 1 [cos (qx) cos (qy) + cos (qx) cos (qz) + cos (qy) cos (qz)] +2η 2 [cos (2qx) + cos (2qy) + cos (2qz)] , n F CC =ñ + 4η 1 cos (qx) cos (qy) cos (qz) + η 2 [cos (2qx) + cos (2qy) + cos (2qz)] , where η 1 and η 2 are the density amplitudes for the first and second reciprocal lattice vectors and q is the unit cell size. Substituting Eqs. (6)(7)(8) into the free energy (4) with D 2 from Eq. (5) and integrating over the elementary cell with size 2π/q eq one gets the free energy for every phase which depends of the equilibrium lattice number q eq . In one-mode approximation, the wave number q eq can be found by minimization of free energy analytically. In the two-mode approximation, obtaining of q eq can be made only numerically. The work [21] proposes the way to minimize the functional by accepting the parameters q eq,BCC = 1/ √ 2 and q eq,F CC = 1/ √ 3 [18][19][20] and to correct the minimization error ∂F/∂q eq | qeq = 0. Asadi and Zaeem [19,21] suggest to introduce the condition for R 0 as: In this way it is possible to prevent the minimization error and incorporate the condition on the R 0 parameter value at the same time. The model of Mkhonta et al. [24] for the multi-mode PFC approximation states parameters b 1 and b 2 as independent. These parameters are similar to the R 0 and R 1 in our model for number of modes N = 2. Therefore, we are calculating structure diagrams using the approaches of Asadi-Zaeem and Mkhonta et al., i.e., we shall use the dependent R 0 by (9) and the independent case for positive and negative values of R 0 . Because interplanar distances are different for different crystal structures we utilize the relevant Q 1,BCC = √ 2, Q 1,F CC = 2/ √ 3 for each phase. For the homogeneous phase, the parameter Q 1 is given by the coexistent crystalline phase. The minimization procedure of F with regard to the density amplitudes η 1 and η 2 is given by the gradient descent algorithm. This  Figure 1. The structure phase diagram for BCC, homogeneous (liquid), FCCstructures at R 1 = 0.05 with the approximation for the R 0 to dispose the wave number minimization error. The parameter Q 1 is taken as different for corresponding phases.
In hightemperature region (small ε = 0..0.15), the stable BCC-structure is absent (due to F F CC < F BCC ). For this set of parameters and approximations, the high temperature transformation of BCC-FCC is absent. The phase transition at much lower temperature T , namely at ε = 0.2, between BCC-and FCC-structures appears in the solid state similarly to the low temperature FCC-BCC phase transitions in a pure iron. Label "H." on the diagram means "homogeneous".
predict the stable structure's coexistence for obtained amplitudes even for small values of ε and negative values of R 0 and R 1 . Using the obtained amplitudes we find the free energy functionals for each phase F homogeneous (ε,ñ, R 0 , R 1 ), F BCC (ε,ñ, R 0 , R 1 ), F F CC (ε,ñ, R 0 , R 1 ). With these free energies, we construct the diagram of crystal structures with their coexistence at values of R 0 = −0.1, 0, 0.1 and R 1 = −0.1, 0, 0.1 for independent R 0 case; and R 1 = 0, 0.05, 0.1 for the dependent parameter R 0 .
4. Construction of structure diagram and discussion 4.1. Structure diagram for three dimensional crystals In this section the structure diagrams are calculated using the solution of the Maxwell area rule, using the chemical potentials and free energy functionals for each phase. The selection mechanism is based on the thermodynamic rule of the minimal energy for a virtually existed structure. Figure 1 shows the domains of existence of the homogeneous (liquid) phase, the crystalline BCC structure and FCC structure in coordinates "normalized average atomic densityñ vs. dimensionless temperature ε" for the dependent parameter R 0 and R 1 = 0.05. We calculate the coexistence separately for the each pair of the structure and then select the structure by its minimal free energy. For the positive R 1 , the BCC region is located in the region of large ε. This region decreases with the decrease of R 1 to the zero. The condition on the parameter R 0 prevents shifting of the BCC structure region under the FCC area. With large negative values of R 0 , the BCC-structure becomes unstable.

4.2.
Structure phase diagram for the independent control parameter R 0 Figure 2 presents the structure diagram for the phases with independent values of R 0 = −0.1 and R 1 = −0.1. We calculated a set of diagrams to find the influence of parameters R 0 and R 1 on the position of lines for coexisted structures. Note that the parameter Q 1 has to be chosen exactly for a given structure to take in account the symmetry of the density profile and the correct free energy of a structure. For the presently constructed diagrams, the overlapping of  Figure 2.
The region of existence of the BCC structure is enlarged relatively to the Fig. 1 and it shows the possibilities of the coexistence of BCC and homogeneous structure. Nevertheless there is no alternation in structures with the increase of ε at the fixedñ. For the complicated sequence of phase transformations in the region of small undercoolings ε = 0 ÷ 0.15, one can expect that FCC-BCC-FCC can appear when the parameter R 0 is free. Label "H." on the diagram means "homogeneous".
regions BCC-FCC-BCC is absent. At small undercooling, i.e. at high temperature the sequence of structures should be BCC-FCC (i.e., high temperature ferrite -austenite in the pure iron). The possible opportunity to obtain such sequence is to introduce in the model an additional atomic interaction (which may lead to appearing of additional structural transformation).
For the positive values of R 0 and R 1 , the BCC region is located in the region of large ε, the lines of coexisted structures do not changed dramatically compared to the Fig. 1. The same situation is observed for the positive R 1 and fixed R 0 . The independent parameter R 0 successfully controls the width of BCC zone and location of the lower part of the region. Simultaneous converging of the parameters to zero leads to widening and getting down of the permitted BCC region. With the positive values of the R 1 there is no strong influence of the R 0 = 0 on the form of the coexisted curves. The negative values of the parameter R 1 allows us to broad the BCC region and to enhance the incline of the "FCC+Homogeneous" line. For independent values of R 0 , there is no limitation for the width of the BCC zone. This leads to appearance of the "Homogeneous(Liquid)-BCC" transition (small ε < 0.02 in Fig. 2). The "FCC+Homogeneous(Liquid)" structure detaches the BCC region following the thermodynamic phase selection rule.

Conclusions
Structure diagrams were calculated for the two-mode PFC model with different control parameters. These parameters are linked with stability of the three dimensional crystal structures.
The parameter R 0 can be dependent that limits the ability to form complex crystal coexistence with small values of the driving force ε. On the other hand, we can reduce the degrees of freedom of the entire system. In such procedure, the reduction of the minimization error is very important.
We have shown the possibilities to find the equilibrium structure diagrams for independent parameter R 0 . With this aim we have used the numerical minimization of the density amplitudes in the dimensionless atomic density expansion. The resulting diagram showed the stable solution for the coexisted curves in the wide region of driving force.
The pair of parameters R 0 and R 1 gives the capability to vary the properties of the material. Despite of the minimization error, it seems to be necessary to separate R 0 and R 1 making