Scale coupling and interface pinning effects in the phase-field-crystal model
Effects of scale coupling between mesoscopic slowly-varying envelopes of liquid-solid profile and the underlying microscopic crystalline structure are studied in the phase-field-crystal (PFC) model. Such scale coupling leads to nonadiabatic corrections to the PFC amplitude equations, the effect of which increases strongly with decreasing system temperature below the melting point. This nonadiabatic amplitude representation is further coarse-grained for the derivation of effective sharp-interface equations of motion in the limit of small but finite interface thickness. We identify a generalized form of the Gibbs-Thomson relation with the incorporation of coupling and pinning effects of the crystalline lattice structure. This generalized interface equation can be reduced to the form of a driven sine-Gordon equation with KPZ nonlinearity, and be combined with other two dynamic equations in the sharp interface limit obeying the conservation condition of atomic number density in a liquid-solid system. A sample application to the study of crystal layer growth is given, and the corresponding analytic solutions showing lattice pinning and depinning effects and two distinct modes of continuous vs. nucleated growth are presented. We also identify the universal scaling behaviors governing the properties of pinning strength, surface tension, interface kinetic coefficient, and activation energy of atomic layer growth, which accommodate all range of liquid-solid interface thickness and different material elastic modulus.
pacs:81.10.Aj, 05.70.Ln, 68.55.A-
Continuum theories have been playing a continuously important role in modeling and understanding a wide range of complex nonequilibrium phenomena during materials growth and processing. For the typical example of liquid-solid front motion and interface growth, sharp-interface or Stefan-type models have been used in early studies to examine various solidification phenomena such as dendritic growth and directional solidification of either pure systems or eutectic alloys Langer (1980). Recent focus has been put on the continuum phase-field approach, which has become a widely-adopted method in materials modeling not only due to its computational advantage as compared to atomistic techniques and also to the complex moving-boundary problems encountered in sharp-interface models, but also due to its vast applicability for a wide variety of material phenomena including solidification, phase transformation, alloy decomposition, nucleation, defects evolution, nanostructure formation, etc. Elder et al. (1994); Karma and Rappel (1998); Elder et al. (2001); Müller and Grant (1999); Kassner et al. (2001); Granasy et al. (2004); Wang and Li (2010).
These continuum methods are well formulated for the description of long wavelength behavior of a system. To incorporate properties related to smaller-scale crystalline details which can have significant impact, additional assumptions or modifications are required. Examples include the consideration of lattice anisotropy for surface tension and kinetics Karma and Rappel (1998), and the incorporation of subsidiary fields describing system elasticity Müller and Grant (1999); Kassner et al. (2001), plasticity Wang and Li (2010), or local crystal orientation Granasy et al. (2004) in phase-field models. However, effects associated with the discreteness of a crystalline system, such as the atomistic feature of lattice growth, are usually absent due to the nature of continuum description. Efforts to partially remedy this in some previous studies include, e.g., the adding of a periodic potential mimicking effects of crystalline lattice along the growth direction in the continuum modeling of surface roughening transition Chui and Weeks (1978); Nozières and Gallet (1987); Hwa et al. (1991); *re:balibar92; *re:mikheev93; *re:rost94; *re:hwa94, although both the form of lattice potential (usually assumed as a sinusoidal function as part of the sine-Gordon Hamiltonian) and the associated parameters were introduced phenomenologically.
More systematic approaches based on some fundamental microscopic-level theories are needed for the construction of continuum field models that incorporate crystalline/atomistic features. One of the recent advances on this front is the development of phase-field-crystal (PFC) methods Elder et al. (2002); *re:elder04; Elder et al. (2007), in which the structure and dynamics of a solid system are described by a continuum local atomic density field that is spatially periodic and of atomistic resolution; thus the small length scale of crystalline lattice structure is intrinsically built into the continuum field description with diffusive dynamic time scales. Both free energy functionals and dynamics of the PFC models can be derived from atomic-scale theory through classical density functional theory of freezing (CDFT) and the corresponding dynamic theory (DDFT), for both single-component and alloy systems Elder et al. (2007); Huang et al. (2010); Greenwood et al. (2010); *re:greenwood11b; van Teeffelen et al. (2009); Jaatinen et al. (2009). Properties associated with crystalline nature of the system, such as elasticity, plasticity, multiple grain orientations, crystal symmetries and anisotropy, are then naturally included, with no additional phenomenological assumptions needed as compared to conventional continuum field theories. This advantage has been verified in a large variety of applications of PFC, ranging from structural, compositional, to nanoscale phenomena for both solid materials Elder et al. (2002); *re:elder04; Elder et al. (2007); Huang et al. (2010); Greenwood et al. (2010); *re:greenwood11b; Jaatinen et al. (2009); Huang and Elder (2008); *re:huang10; Wu and Voorhees (2009); Spatschek and Karma (2010); Muralidharan and Haataja (2010); Berry and Grant (2011); Elder et al. (2012) and soft matters van Teeffelen et al. (2009); Wittkowski et al. (2011).
An important feature of the PFC methodology is the multiple scale description it provides, as can be seen from its amplitude representation. The system dynamics is described by the behavior of “slow”-scale (mesoscopic) amplitudes/envelopes of the underlying crystalline lattice, as a result of the amplitude expansion of PFC density field in either pure liquid-solid systems Goldenfeld et al. (2005); *re:athreya06; Yeon et al. (2010) or binary alloys Elder et al. (2010); Huang et al. (2010); Spatschek and Karma (2010). Note that in these amplitude equation studies, although most lattice effects have been incorporated in the variation of complex amplitudes (mainly via their phase dynamics Huang and Elder (2008); *re:huang10), the spatial scales of the mesoscopic amplitudes and the microscopic lattice structure are assumed to be separated (i.e., the assumption of “adiabatic” expansion). However, this assumption only holds in the region of slowly varying density profile either close to the bulk state or for diffuse interfaces, and hence is valid only at high enough system temperature. In low or intermediate temperature regime showing sharp liquid-solid or grain-grain interfaces, amplitude variation around the interface would be of order close to the lattice periodicity; thus the two scales of amplitudes vs. lattice can no longer be separated, resulting in the “nonadiabatic” effect due to their coupling and interaction. Such scale coupling leads to an important effect of lattice pinning that plays a pivotal role in material growth and evolution, as first discussed by Pomeau Pomeau (1986) and later demonstrated in the phenomena of fluid convection and pattern formation Bensimon et al. (1988); Boyer and Viñals (2002a); *re:boyer02b. To our knowledge, these scale coupling effects have not been addressed explicitly in all previous phase-field and PFC studies of solidification and crystal growth.
In this paper we aim to identify these coupling effects between mesoscale structural amplitudes and the underlying microscopic spatial scale of crystalline structure, via deriving the nonadiabatic amplitude representation of the PFC model. What we study here is the simplest PFC system: two-dimensional (2D), single-component, and of hexagonal crystalline symmetry, as our main focus is on examining the fundamental aspects of scale coupling and bridging that are missing in previous research, and also on further completing the multi-scale features of the PFC methodology. The explicit expression of the resulting pinning force during liquid-solid interface motion, and also its scaling behavior with respect to the interface thickness, are determined in this work, through the application of sharp/thin interface approach (given finite interface thickness) to the amplitude equations. This leads to a new set of interface equations of motion, in particular a generalized Gibbs-Thomson relation that incorporates the pinning term and also its reduced form of a driven sine-Gordon equation. The pinning of the interface to the underlying crystalline lattice structure, and the associated nonactivated vs. nucleated growth modes, can be determined from analytic solutions of the interface equations for the case of planar layer growth.
Ii Nonadiabatic coupling in amplitude equations
In the PFC model for single-component systems, the dynamics of a rescaled atomic number density field is described in a dimensionless form Elder et al. (2002); *re:elder04; Elder et al. (2007); Huang et al. (2010)
where measures the temperature distance from the melting point, with proportional to the bulk modulus, and we have after rescaling over a length scale of lattice spacing. The noise field has zero mean and obeys the correlations
with , where is a rescaled constant depending on and Huang et al. (2010), and is the system temperature.
To derive the corresponding 2D amplitude equations in the limit of small , we need to first distinguish the “slow” spatial and temporal scales for the amplitudes/envelopes of the structural profile, i.e.,
from the “fast” scales of the underlying hexagonal crystalline structure. We then expand the PFC model equation (1) based on this scale separation and also on a hybrid approach combining the standard multiple-scale expansionManneville (1990); Cross and Hohenberg (1993) and the idea of “Quick-and-Dirty” renormalization group method Goldenfeld et al. (2005); *re:athreya06 (see Ref. Huang et al. (2010) for details). To incorporate the coupling between these “slow” and “fast” scales, which leads to nonadiabatic corrections to the amplitude equations, we use an approach based on that given in Refs. Bensimon et al. (1988); Boyer and Viñals (2002a); *re:boyer02b which address front motion and locking in periodic pattern formation during fluid convection.
where are the three basic wave vectors for 2D hexagonal structure (i.e., the 3 “fast”-scale base modes)
and the slow scaled fields, including (complex amplitudes of mode ) and (real amplitude of the zero wavenumber neutral mode as a result of PFC conserved dynamics), are represented as power series of : , . Note that in Eq. (4) higher harmonic terms have been neglected.
where is the linear operator in PFC, refers to the slow-scale expansion to all orders of , and , , and are slow operators. In Eq. (6), is the slow-scale correspondence of the effective free energy given below [with replaced by and replaced by ]:
which is the same as the previous amplitude expansion result Yeon et al. (2010); Huang and Elder (2008); *re:huang10; Huang et al. (2010); also for other variables () and () related to higher harmonics,
where and (with the atomic lattice spacing for the hexagonal/triangular structure and , as illustrated in Fig. 1), which are the atomic spatial periods along the and directions respectively. Note that Eq. (9) can be also viewed as the combination of the solvability conditions obtained at all different orders of in multiple-scale expansion Huang et al. (2010).
In the limit of , i.e., close to the melting temperature, the spatial variation of amplitudes and is of much larger scale compared to the atomic lattice variation scales and . Thus “slow” and “fast” length scales in the integral of Eq. (9) can be separated as in standard multiple-scale analysis (i.e., and be treated as independent variables), leading to the Ginzburg-Landau-type amplitude equations obtained in previous studies Yeon et al. (2010); Huang and Elder (2008):
Note that to derive Eq. (10) the long-wavelength approximation has been used as before.
However, such assumption of scale separation would not hold when is of larger value (still small but finite, corresponding to low/moderate material temperature far enough from the melting point). Although the amplitudes/envelopes still vary slowly in the bulk, the interface, either between liquid and solid states or between different grains, could be thin or sharp, with its width comparable to “fast” lattice scales (e.g., of few lattice spacings). Thus functions of and in Eq. (6) can no longer be decoupled from the atomic-scale oscillatory terms (with and integers and ) in the integration of in Eq. (9). Using an approximation similar to that in Ref. Boyer and Viñals (2002a); *re:boyer02b, we only keep the lowest-order coupling terms, i.e., terms coupled to lowest modes (with largest atomic lengths) along both and directions, including , corresponding to atomic layer spacing , and , with atomic layer spacing . Couplings to higher modes, such as (with length scale ), (with length scale ), etc., are neglected. The following scale-coupled amplitude equations can then be derived from Eqs. (9) and (6):
with , , and if assuming equal contribution from all eigenmodes . In the above generalized amplitude equations (11)–(14), the integration terms explicitly yield the coupling between “slow” (for structural amplitudes or envelopes) and “fast” (for atomic lattice variations) spatial scales, that is, the nonadiabatic corrections. The first 3 coupling terms in each of Eqs. (11)–(13) for the dynamics of complex amplitudes are associated with lattice modes of length scale (), while the other 4 terms correspond to the coupling to modes with length scale (). The nonadiabatic effect for dynamics is weaker, with only couplings to the length scale given in Eq. (14). Note that in the bulk state of single crystal or homogeneous liquid, the amplitude functions , constants, and hence all the integrals in Eqs. (11)–(13) are equal to zero; we can then recover the original amplitude equations (10) without any nonadiabatic coupling, as expected.
Iii Interface equations of motion with lattice pinning
To illustrate the important effects of scale coupling identified in above nonadiabatic amplitude equations, here we consider a system of coexisting liquid and solid phases, with the average interface normal direction pointed along . The extension of our derivation and results to other interface orientations is straightforward.
(: for and ; : for ), where is a chemical potential of the system, and is the bulk free energy density given from Eq. (7):
To derive the corresponding equations of motion for the interface with finite thickness (i.e., in the sharp/thin interface limit Karma and Rappel (1998); Elder et al. (2001)), we follow the general approach developed by Elder et al. Elder et al. (2001) with the use of projection operator method. In this approach, a small parameter is introduced, which represents the role of the interface Péclet number, and the system is partitioned into two regions: an inner region around the interface, defined by , and an outer region far from the interface (i.e., ), where length scales as , and is the component of a local curvilinear coordinate in the interface normal direction. In this curvilinear coordinate , the two orthogonal unit vectors are defined as (local normal of the interface) and (tangent to the interface), where is the angle between and the axis; also
with the local curvature .
iii.1 Outer equations
In the outer region which is far enough from the interface and close to the bulk states, the scale coupling term in Eq. (16) can be neglected, and the slowly varying amplitude fields and depend on rescaled spatial variables and rescaled time . Expanding the outer solution of amplitudes in powers of , i.e.,
and at ,
where “” refers to replacing by in the derivative or , and “” refers to the corresponding results up to 1st order of and . Note that if assuming the system to be not far from a liquid-solid equilibrium state, Eq. (21) of yields the bulk equilibrium solutions of the uniform liquid () or solid () state
with the corresponding equilibrium chemical potential .
iii.2 Inner expansion and lattice coupling effect
For the inner region (), the amplitudes and chemical potential can be also expanded as
Due to the presence of interface at , the amplitudes are expected to vary rapidly along the normal direction but slowly along the arclength of the interface, leading to the rescaling . Considering small interface fluctuations and noise amplitude, we assume that , , , and . Thus from Eq. (19) we have , , and , where for ,
and for ,
To address the time relaxation of system in the inner region, as usual we use a coordinate frame co-moving with the interface at a normal velocity , and hence . The inner expansion of the nonadiabatic amplitude equations (16) and (17) can then be given by: For ,
giving the equilibrium chemical potential ; At ,