Suppression of coarsening and emergence of oscillatory behavior in a Cahn-Hilliard model with nonvariational coupling
SSuppression of coarsening and pattern formation in a two-fieldCahn-Hilliard model with nonvariational coupling
Tobias Frohoff-H¨ulsmann, ∗ Jana Wrembel, and Uwe Thiele
1, 2, 3, † Institut f¨ur Theoretische Physik, Westf¨alische Wilhelms-Universit¨at M¨unster,Wilhelm-Klemm-Str. 9, 48149 M¨unster, Germany Center for Nonlinear Science (CeNoS),Westf¨alische Wilhelms-Universit¨at M¨unster,Corrensstr. 2, 48149 M¨unster, Germany Center for Multiscale Theory and Computation (CMTC),Westf¨alische Wilhelms-Universit¨at, Corrensstr. 40, 48149 M¨unster, Germany
Abstract
We investigate a generic model for passive and active ternary mixtures consisting of a two-field Cahn-Hilliard model with respective variational and nonvariational coupling between the fields, i.e., the latterbreaks the gradient dynamics structure of the model. Already a linear stability analysis of the homogeneousmixed state shows that activity allows not only for the usual large-scale stationary instability known fromthe passive case but also for small-scale stationary and large-scale oscillatory instabilities. In consequenceof the small-scale instability, a linear suppression of the usual Cahn-Hilliard coarsening dynamics occurs.Subsequently, we provide an extensive nonlinear analysis, first, of the passive case where we discuss thephase behavior in the thermodynamic limit and relate it to appropriate bifurcation diagrams for systems offinite size. Second we focus on the active case where bifurcation diagrams and selected direct time sim-ulations allow us to discuss arrest and complete suppression of coarsening, the emergence of drifting andoscillatory states, and the behavior in a number of relevant limiting cases. Throughout the work we em-phasize the relevance of conservation laws and related symmetries for the encountered intricate bifurcationbehavior. ∗ Electronic address: t [email protected]; ORCID ID: 0000-0002-5589-9397 † Electronic address: [email protected]; URL: ; ORCID ID: 0000-0001-7989-9271 Preprint– contact: t [email protected] – October 5, 2020 a r X i v : . [ c ond - m a t . s o f t ] O c t hase separation, also called demixing, unmixing or decomposition is a universal process oc-curring in many experimental systems where an initially homogeneous mixed state decomposesinto different phases [1–3]. If quenched into a linearly unstable state, phase heterogeneities de-velop on a typical lengthscale determined by the quench. Over time, the developing structures con-tinuously coarsen, i.e., their average size increases and their number decreases [1]. The simplestdynamical model for such processes is the Cahn-Hilliard (CH) equation, a nonlinear, dissipativemodel originally proposed to describe the dynamics of demixing of isotropic solid or fluid binarysolutions [4, 5]. Extensions to decomposing mixtures of multiple components are also available[6, 7]. Already in the case of a binary mixture, the generic CH model captures many qualitativefeatures of demixing and thus is widely applied from material science to soft matter. Variants andextensions are also increasingly used in biophysical contexts. Examples include descriptions ofprotein patterns near membranes of living cells [8, 9], of the motility-induced phase separation ofactive Brownian particles [10–13], and of the suppression of Ostwald ripening in active emulsionsrelevant for centrosome dynamics in biological cells [14–16].A common feature of most variants of CH models outside the biophysical context is that thedescribed dynamics of a concentration or density field φ ( x, t ) conserves a mass-like quantity and results in the decrease of an underlying energy F [ φ ] . These physical properties directly determinethe form of the equation: a conservation law with a variational form. With other words, theCH model represents a mass-conserving gradient dynamics that describes the transition from an(unstable) initial state to a (stable or metastable) equilibrium state that minimizes F . The finalstate is not necessarily the global energy minimum. If it is the global minimum, it corresponds tothe thermodynamic equilibrium only in the thermodynamic limit, i.e., for diverging system size.For a discussion how this limit is approached with increasing system size see Ref. [17].If the system boundaries do not sustain any throughflow, and no energy is fed into the systemin other ways, e.g., by chemical reactions, we call the system “passive”. This, together withthe variational form implies that no time-periodic states can occur and, in particular, all linearmodes are stationary. However, there exist several settings where the system becomes “driven” or“active”. One option is the addition of a lateral driving force in combination with a correspondingflux of material across the system boundaries. The resulting convective CH equation is studied,e.g., in [18–21]. In this case, the driving term breaks the parity symmetry of the CH equation, i.e.,in a one-dimensional (1D) system the left-right symmetry. This implies that all states travel. Thiscase shall not concern us here. 2 Preprint– contact: t [email protected] – October 5, 2020 nother option is to add “activity”, normally, corresponding to additional terms that do notbreak the parity symmetry but are nevertheless nonvariational, i.e., they break the gradient dynam-ics structure of the equation. Often, such contributions result from a chemo-mechanical coupling,e.g., for self-propelled constituents, and indicate that the system acquires energy from outsidethat is then dissipated within. An example is an active CH-type equation that describes phaseseparation processes in nonequilibrium systems. It models aspects of the so-called “active phaseseparation” in suspensions of self-propelled particles [11, 22, 23], is also relevant in the context ofcell polarization and chemotactic aggregation [24, 25], and can close to the corresponding criticalpoint be systematically derived as leading (passive CH equation) and next-to-leading order (activeextensions) dynamics [13, 23]. Despite its nonvariational character, generalized thermodynamicquantities can be defined such as nonequilibrium pressure and chemical potential which resultin nonequilibrium coexistence conditions and an “uncommon tangent (Maxwell) construction”[10, 26]. Other active CH-type equations do not allow for the definition of such generalized ther-modynamic quantities. For systems of more than one dimension a term can be added that supportscirculating currents [27].In the context of applications, biophysical and other, often several degrees of freedom areinvolved, i.e., dynamic models describe the coupled evolution of several density- or concentration-like order parameter fields that each may follow a conserved or nonconserved dynamics. Again,models can be variational or nonvariational. In the former case, such models describe, e.g., phaseseparation in ternary [6, 7] and multicomponent [28–30] mixtures including in membranes (seemodel I in [9]). Also thin-film models for layers of solutions and suspensions [31, 32] belong tothe same class of equations. In the nonvariational case, typical examples are models for phaseseparation in ternary mixtures with chemical reactions [33, 34], membrane models that includechemical reactions [9, 35], and thin-film models for active liquids [36]. Such membrane modelsconsist, e.g., of reaction-diffusion (RD) equations for three fields with one conservation law (seemodel II in [9] and [37]), and of four fields with CH or RD dynamics with two conservation laws[35]. A five-field model with two conservation laws is considered in [38] where also a simplerconceptual model is analyzed consisting of a two-field RD system with one conservation law.An active emulsion model describing, e.g., centrosome dynamics in biological cells, employsa reactive coupling of two CH equations keeping only one overall conservation law [14, 16].A “non-reciprocal CH model” consisting of two CH equations with nonvariational coupling isinvestigated in Refs. [39, 40] as a description of interacting scalar active particles where both3
Preprint– contact: t [email protected] – October 5, 2020 pecies are individually conserved. It shows demixing at small non-reciprocal coupling whichtransitions to oscillatory behavior at high activity, e.g., self-propelled globally ordered bands. Inanother conceptual model, two CH equations, i.e., two conservation laws, are coupled in a way thatbreaks both conservation laws and the variational structure [41]. It is found that the coupling canarrest the coarsening process typical for CH dynamics and may even result in oscillatory dynamics.A central feature of phase separation as modeled by the CH model is the already mentionedcoarsening that results in a continuous increase of typical sizes of the developing phase-separatedregions, i.e., drops/clusters, holes or labyrinthine structures [1]. Coarsening proceeds through thetwo main modes of volume transfer (known as Ostwald ripening) and by translation (known ascoalescence). The volume transfer mode moves material between structures without moving theircenters, i.e., their sizes change. In contrast, the translation mode moves the structures withoutchanging their sizes. More details on coarsening behavior in the CH equation and related thin-filmequations are, e.g., given in [3, 42–44].Coarsening may be suppressed by heterogeneities in the (still variational) system, e.g., fordrops on a substrate with wettability patterns [45] or phase separation in a spatially modulatedtemperature profile [46]. In diblock copolymer melts described by a single CH equation withlong-range interactions (Oono-Shiwa model) the system is stabilized at a certain length scale [47].Coarsening can also be suppressed by driving or activity. Studies of the convective CH equation[18, 48] show that an increase in the lateral driving force results in a transition towards chaoticwaves. This implies that there exist parameter regions where driving arrests coarsening [49].Aspects of the resulting bifurcation structure are presented in [21].In most active one-field CH models employed to describe motility-induced phase separation,coarsening closely resembles its counterpart in the standard passive model [11]. However, “re-verse Ostwald ripening” for vapor bubbles and liquid clusters is described for an active CH modelin two dimensions with two types of nonvariational contributions: a nonequilibrium chemical po-tential and a nonequilibrium flux, itself related to a nonlocal chemical potential [27]. Suppressionof coarsening is also observed for active models involving coupled CH equations. Reference [41]shows that suppression occurs already at weak nonvariational coupling between the two concen-tration fields. It is argued that each structured field acts as heterogeneity for the other one and theresulting pinning arrests coarsening. Linear stability analysis and direct time simulations showthat besides the arrest of coarsening, the nonvariational coupling may also induce the structuresto travel or oscillate. With other words the chosen coupling dramatically changes central features4
Preprint– contact: t [email protected] – October 5, 2020 f the phase separation model. Similar phenomena are also observed in more complex models forreactive decomposition [16, 34, 35].Motivated by these various phenomena in active phase-separating systems, here we study a sys-tem of generic kinetic equations consisting of two coupled CH equations where the coupling (i)maintains both conservation laws and (ii) consists of clearly separated variational (reciprocal) andnonvariational (non-reciprocal) contributions. This allows us to analyze the qualitative transitionsin the dynamics of two conserved quantities that occur when going from a variational to a non-variational model. The nonvariational case has already been studied in Ref. [39] (non-reciprocalCH model) and with some simplification in Ref. [40] with a focus on the emergence of travelingstates. Here, we systematically show that the nonvariationally coupled CH model exhibits a muchricher selection of phenomena. Especially, our analysis allows for a deeper understanding of sim-ilarities and differences between Ref. [39] and the study in Ref. [41] where the coupling does notmaintain the conservation property and is purely nonvariational. As a result it shall be possibleto identify features of related system-specific models in the literature as generic features resultingfrom conservation laws. In particular, we show that for such systems a nonvariational couplingcan result in linear and nonlinear arrest and suppression of coarsening. Our analysis explains whythis behavior can not be observed in the seemingly identical model studied in Refs. [39, 40].Our work is structured as follows. In Section I we introduce the model and discuss our nu-merical approach. Subsequently, Section II provides a linear stability analysis of the uniform statein the variational and the nonvariational case. For the latter, we discuss the transition from large-to small-scale stationary instabilities and the occurrence of a large-scale oscillatory instability.Relying on these linear considerations, in sections III and IV we employ numerical bifurcationanalysis to study the nonlinear behavior for the variational and nonvariational case, respectively.Subsequently, we investigate how coarsening can be arrested or fully suppressed (section IV A)and how structures start to move (section IV C) when the nonvariational coupling is increased. Thebifurcation study is accompanied by selected direct numerical simulations. Section V concludeswith a summary and outlook.
I. GOVERNING EQUATIONS
The classic Cahn-Hilliard model describes the dynamics of diffusive phase decomposition pro-cesses in various (solid-solid, liquid-liquid, liquid-gas) demixing processes of binary systems. For5
Preprint– contact: t [email protected] – October 5, 2020 scalar order parameter field φ ( x , t ) the corresponding conserved gradient dynamics reads ∂φ∂t = ∇ · (cid:20) Q ( φ ) ∇ δ F [ φ ] δφ (cid:21) (1)where Q ( φ ) is a positive definite mobility function (or constant) and F [ φ ( x , t )] = (cid:90) V (cid:104) κ |∇ φ | + f ( φ ) (cid:105) d x (2)is the underlying free energy. It consists of an interface contribution in the form of a square-gradient term with interface stiffness κ > and a bulk contribution, that is often approximated bythe simple potential f ( φ ) = a φ + b φ . (3)Here, b > and either a > (case of single minimum) or a < (double-well potential). Notethat Eq. (1) is parity and field-inversion symmetric, i.e., it does not change its form for x → − x and φ → − φ , respectively.The variation of the energy δ F /δφ corresponds to a chemical potential µ and Eq. (1) cancompactly be written as continuity equation ∂ t φ + ∇ · j = 0 with the flux j = − Q ∇ µ . The energymonotonically decreases in time (see, e.g., [50]), i.e., it is a passive system.For a < there exists a φ -range of unstable uniform states that eventually develop into aphase-separated state. In the thermodynamic limit of an infinite system, the interface contributionin Eq. (2) can be neglected and the two coexisting phases (as obtained by a Maxwell construction)exactly correspond to the minima of f ( φ ) as they have identical chemical potential and pres-sure. For a detailed discussion how this thermodynamic picture relates to bifurcation diagramsof steady states for finite-size systems where interface contribution can not be neglected see therecent Ref. [17].After this brief revision of the classic one-field CH model, we next introduce the coupled systemof two CH equations studied here. Without coupling, each of the two equations corresponds toEq. (1), though with different constants, and the simple coupling is chosen in such a way thatit respects the field inversion symmetry ( φ , φ ) → ( − φ , − φ ) . After restriction to one spatialdimension and nondimensionalisation (explained in appendix A) the kinetic equations are ∂φ ∂t = 1 L ∂ ∂x (cid:18) − L ∂ φ ∂x + f (cid:48) ( φ ) − ( ρ + α ) φ (cid:19) ∂φ ∂t = QL ∂ ∂x (cid:18) − κL ∂ φ ∂x + f (cid:48) ( φ ) − ( ρ − α ) φ (cid:19) . (4)6 Preprint– contact: t [email protected] – October 5, 2020 ith f = aφ / φ / and f = ( a + a ∆ ) φ / φ / . Note that the dynamics of each field isconserved, i.e., at all times (cid:90) d xφ = ¯ φ (cid:90) d xφ = ¯ φ . (5)where the ¯ φ i are parameters set by the initial conditions. Note that the field inversion symmetrydoes not normally hold for the deviations φ i − ¯ φ i that are often the relevant quantities to consider.The other parameters are the nondimensional domain size L , mobility ratio Q , effective tempera-ture a , temperature shift a ∆ , and ratio of interface rigidities κ . Note that the parameter L allowsus to easily change the physical domain size while keeping the computational domain size fixedto one. The respective final terms in Eqs. (4) represent the coupling. It contains a variational partof strength ρ and a nonvariational part of strength α . Increasing or decreasing α from the passivereference case ( α = 0 ) one can investigate the system behavior with increasing activity.In the passive case ( α = 0 ), the governing equations (4) are of simple gradient dynamics form ∂ t φ i = ∂ x ( Q i ∂ x δ F /δφ i ) with i = 1 , . The energy F ( φ , φ ) = F ( φ ) + F ( φ ) + F c ( φ , φ ) (6)is the sum of the two energies F and F for the decoupled fields, that are of the form (2) and thecoupling contribution F c = − ρφ φ . The coupling in the passive case is purely energetic. Notethat we do not consider dynamic coupling as encoded in a mobility matrix, e.g., we exclude cross-diffusion. For a discussion of some such systems see [32]. The chosen active coupling representsone way to break the variational form of the passive case. We mainly investigate steady states andtime-periodic states in a spatial domain with periodic boundaries by numerical path continuationfor which we use the Matlab package pde2path [51, 52], accompanied by selected time simula-tions. Numerical path continuation is a tool which allows us to track linearly stable and unstablesteady states while varying a primary control parameter. Beginning with a starting steady stateat some parameter value, pde2path applies tangent predictors and Newton correctors to convergeinto a steady state at neighboring parameter values. Especially pseudo-arclength continuation is aparametrization which allows for reversals in the direction of control parameter steps – a featureessential to track solution branches through folds. For steady states without mean flow, we can7
Preprint– contact: t [email protected] – October 5, 2020 ntegrate Eq. (4) twice to obtain − L ∂ φ ∂x + f (cid:48) ( φ ) − ( ρ + α ) φ + µ − κL ∂ φ ∂x + f (cid:48) ( φ ) − ( ρ − α ) φ + µ , (7)where the integration constants µ i are nonequilibrium “chemical potentials”. To impose the con-servation of both fields, we expand Eq. (7) by adding the constraints (5). In consequence, duringcontinuation runs using Eqs. (7) the µ i are added as secondary control parameter. When usingEqs. (4) with ∂ t φ i = 0 the role is taken by the strengths of additional “virtual” source terms, thatare then kept at zero. Furthermore, linear stability of steady states is determined and, hence, allkinds of local bifurcations are detected. This allows one to switch to other steady state branches.Branches of time-periodic states are also continued in Sec. IV C and the necessary extension ofthe continuation technique can be found in [52]. To present the resulting bifurcation behavior,normally the norm || δφ || ≡ (cid:115)(cid:90) (cid:88) i =1 , (cid:0) φ i − ¯ φ i (cid:1) d x (8)is employed as solution measure. The effective temperature a is used as primary control parameterexcept for part of Sec. III. II. LINEAR STABILITY OF HOMOGENEOUS STATE
First, we analyze the linear stability of the homogeneous states. Due to mass conservation,any steady homogeneous state φ ( x ) ≡ ( φ ( x ) , φ ( x )) = ( ¯ φ , ¯ φ ) = ¯ φ solves Eqs. (4). For theperturbation we employ the harmonic ansatz φ ( x, t ) = ¯ φ + δ (cid:101) φ e ikx + λt , (9)introduce it in Eqs. (4) and linearize in δ (cid:28) . The obtained linear algebraic system is λ (cid:101) φ = − (cid:18) kL (cid:19) (cid:0) kL (cid:1) + f (cid:48)(cid:48) − ( ρ + α ) − Q ( ρ − α ) Q (cid:16) κ (cid:0) kL (cid:1) + f (cid:48)(cid:48) (cid:17) (cid:101) φ ≡ − (cid:18) kL (cid:19) B (cid:101) φ (10)rewritten as ˜ λ φ = − B φ (11)8 Preprint– contact: t [email protected] – October 5, 2020 ith ˜ λ = λ/q and q = k/L . The resulting dispersion relations are ˜ λ ± = 12 (cid:104) − tr B ± (cid:112) (tr B ) − B (cid:105) (12)with tr B = q (1 + Qκ ) + f (cid:48)(cid:48) + Qf (cid:48)(cid:48) and det B = Q (cid:2) q + f (cid:48)(cid:48) (cid:3) (cid:2) κq + f (cid:48)(cid:48) (cid:3) + Q ∆ . where we defined the difference in coupling strengths ∆ ≡ α − ρ . The rescaled eigenvaluesEq. (12) are of the same form as those obtained for two coupled reaction diffusion systems [53].The original eigenvalues λ are obtained by multiplying Eq. (12) with k /L which ensures theconservation of both fields.In the following we use f (cid:48)(cid:48) and f (cid:48)(cid:48) as primary and secondary control parameter, respectively.Analyzing Eq. (12) gives us conditions for three different primary instabilities: (i) large-scaleoscillatory (Hopf) instability (ii) small-scale stationary (Turing) instability and (iii) large-scalestationary (Cahn-Hilliard) instability. In the Cross-Hohenberg classification they are termed (i)type II o , (ii) type I s , and (iii) type II s instability [54]. Large-scale [small-scale] instabilities are alsocommonly termed long-wave [short-wave] instabilities. Note that only instability (iii) occurs inthe uncoupled CH equations. We will show that | α | > | ρ | is a necessary condition for instabilities(i) and (ii) to occur. (i) First we consider the large-scale oscillatory instability related to Hopf bifurcations. Theonset of oscillatory instabilities is characterized by λ ± ,c = ± iω c , i.e., with Eq. (12) this requires tr B = 0 ⇒ f (cid:48)(cid:48) = − (1 + Qκ ) q − Qf (cid:48)(cid:48) . (13)Since Q, κ > , the largest f (cid:48)(cid:48) always occurs at q = q c = 0 , i.e., here the oscillatory instabilitiesare always large-scale. Therefore, the Hopf-threshold is f (cid:48)(cid:48) H = − Qf (cid:48)(cid:48) . (14)However, both eigenvalues of the original linear system [Eq. (10)] remain real and zero at exactly k = 0 due to the conservation properties, i.e., the critical frequency is ω c = q c ˜ ω c = 0 with ˜ ω c = det B (cid:12)(cid:12) q = q c = (cid:112) Q (cid:113) − Qf (cid:48)(cid:48) + ∆ (15)In summary, the large-scale oscillatory instability occurs if ∆ > Qf (cid:48)(cid:48) at f (cid:48)(cid:48) = f (cid:48)(cid:48) H . Especially,if the two coupled subsystems are identical ( f (cid:48)(cid:48) = f (cid:48)(cid:48) ) the Hopf-threshold is at f (cid:48)(cid:48) H = 0 . This9 Preprint– contact: t [email protected] – October 5, 2020 mplies that for identical subsystems with purely nonvariational coupling ( ρ = 0 ), oscillatorybehavior occurs at arbitrarily small nonvariational coupling α . Appendix D focuses on this specialcase. However, a stronger contrast between the two coupled systems implies that a larger coupling | α | is needed to obtain oscillatory behavior. (ii) Next we consider the small-scale stationary instability related to Turing bifurcations. Itoccurs if ˜ λ + , c = 0 at q c (cid:54) = 0 and requires det B = 0 ⇒ f (cid:48)(cid:48) = − ∆ f (cid:48)(cid:48) + κq − q (16)The maximum of f (cid:48)(cid:48) ( q ) is obtained via d f (cid:48)(cid:48) / d q = 0 and yields the critical wavelength q c = 1 κ (cid:104) ±√ κ ∆ − f (cid:48)(cid:48) (cid:105) (17)For κ > [ κ < ] the plus [minus] sign in Eq. (17) corresponds to a maximum, the minus[plus] sign to a minimum in the dispersion relation, the latter not being relevant for the onset ofinstability. That is, for a Turing instability we demand q c > ⇒ f (cid:48)(cid:48) < ±√ κ ∆ for κ ≷ (18)In particular, it requires nonvariational coupling stronger than the variational one, i.e. | α | > | ρ | ,otherwise the root becomes complex. For comparison with other studies (see conclusion) it isimportant to note that for κ = 1 and κ = 0 no small-scale stationary instability is possible.Inserting q c in (16) gives the Turing instability threshold f (cid:48)(cid:48) T = 1 κ (cid:104) f (cid:48)(cid:48) ∓ √ κ ∆ (cid:105) (19)relevant for the related pitchfork bifurcations. (iii) Finally, we consider the large-scale stationary instability, i.e., the common case of theclassical one-field CH equation. It is characterized by λ + , c = 0 at q c = 0 and occurs at f (cid:48)(cid:48) CH = − ∆ f (cid:48)(cid:48) . (20)The related bifurcations are again pitchfork bifurcations.We see that parameters mobility ratio Q , rigidity ratio κ and the difference in couplingstrengths ∆ determine the three instability thresholds [cf. Eqs. (14),(19),(20)]. Figure 1 providesa qualitative overview of the linear stability behavior in the ( f (cid:48)(cid:48) , f (cid:48)(cid:48) ) -plane. Hopf- (14), Turing-(19) and CH- (20) instability thresholds are given by blue, orange and green lines, respectively.10 Preprint– contact: t [email protected] – October 5, 2020 − f − − − f (a) s t a b l e HopfTuringCH − − f − − − f (b) s t a b l e − − f − − − f (c) s t a b l e FIG. 1:
Linear stability diagrams in the ( f (cid:48)(cid:48) , f (cid:48)(cid:48) ) -plane show thresholds of Hopf- [Eq. (14) ],Turing-[Eq. (19) ] and Cahn-Hilliard-instabilities [Eq. (20) ] with blue, orange and green lines, respectively, fordifferent values of κ and ∆ = α − ρ at fixed Q = 1 . The boundary of the linearly stable region [upperright corner] is marked by heavy solid lines. The thin solid lines indicate where further instabilities set inbeyond the dominating one. The dashed orange line indicates where the minimum of a dispersion relationof Turing type passes zero. The black dashed lines indicate the stability boundary of an uncoupled system(or for identical coupling strengths, i.e. for ∆ = 0 ). Panel (a) is for positive ∆ = 0 . and κ = 0 . < ,panel (b) is for κ = 3 . > and ∆ = 0 . > fixed. The square symbol marks the codimension-2point [Eq. (21) ] where Hopf- and Turing instabilities occur simultaneously. Panel (c) represents the case of | ρ | > | α | with ∆ = − . and κ = 3 . where only CH-instability exists. Preprint– contact: t [email protected] – October 5, 2020 he linearly stable region is in the upper right corner its boundary marked by heavy colored linesthat represents the onset of the different instabilities. For reference, dashed black lines indicate theinstability thresholds for the CH instability of a uncoupled system (also valid at general ∆ = 0 ).Panel (a) and (b) show stability diagrams for positive ∆ , where Hopf-, Turing-, and CH-instabilities occur while in panel (c) for ∆ < only CH-instabilities exist. Further comparisonreveals that the stable region widens [shrinks] for increasing [decreasing] ∆ . Hence, especiallythe purely variational coupling ρ always acts destabilizing. The CH-instability thresholds (greenlines) are hyperbolas [cf. Eq. (20)] which flip quadrants when ∆ changes sign. There are twoTuring-instability thresholds (orange lines) resulting from the two signs in Eq. (19). For κ < [panel (a)] the upper line corresponding to the plus sign refers to a maximum in the dispersionrelation, thus, is relevant for the stability boundary (heavy orange line), whereas the lower line isrelated to a minimum (dotted orange line). In contrast for κ > [panel (b)], the lower orange linematters. In both cases, the relevant Turing line crosses the Hopf line. The crossing point (blackfilled square) marks a codimension-2 point where both instabilities have their onset at same valueof the primary control parameter f (cid:48)(cid:48) . This requires adjustment of a second control parameter, here f (cid:48)(cid:48) = 2 ±√ κ ∆1 + Qκ = − f (cid:48)(cid:48) Q . (21)The Turing lines terminate where they tangentially approach the CH lines at f (cid:48)(cid:48) T end = ±√ κ ∆ and the critical wavenumber reaches zero. The Hopf lines also end on the CH lines where ˜ ω c becomes zero at f (cid:48)(cid:48) H end = ± (cid:113) ∆ Q . The two end points mark the transition from Turing- and Hopf-instability to CH-instability, respectively. They do not correspond to a coexistence of differentlinear instabilities as does the codimension-2 point. Especially, in the particular non-generic case κ = Q = 1 one has f (cid:48)(cid:48) = f (cid:48)(cid:48) T end = f (cid:48)(cid:48) H end , (22)and all three special points coincide. It is remarkable that in this case the Turing lines completelydisappear since the eigenvalues become complex at the threshold implying that no small-scalestationary instability occurs at all (not shown).Up to here, we have discussed the linear stability behavior of the model Eq. (4) for arbitrary f (cid:48)(cid:48) and f (cid:48)(cid:48) . In the following we focus on our specific case with f (cid:48)(cid:48) = a + 3 ¯ φ , f (cid:48)(cid:48) = a + a ∆ + 3 ¯ φ and Q = 1 . We discuss the resulting dispersion relations and stability boundaries for the passive(Sec. II A) and active (Sec. II B) case. Then the effective temperature a is employed as maincontrol parameter corresponding to diagonal cuts through the stability diagrams in Fig. 1. Some12 Preprint– contact: t [email protected] – October 5, 2020 urther details of our specific case deduced from the general stability behavior are presented inappendix B.
A. Passive system k − k + k − . − . − . − . − . . . . λ n =1 n =2 (a) n =1 n =2 k − . − . . . . a s t a b l e unstable(b) ρ =1.0 ρ =0.5 ρ =0.0 FIG. 2:
Linear stability behavior of homogeneous states for two passively coupled Cahn-Hilliard equa-tions, i.e., in the variational case ( α = 0 ). Panel (a) shows the dispersion relations λ ± ( k ) [Eq. (B1) ] atvariational coupling strength ρ = 0 . beyond the onset of the large-scale linear instability, i.e., here for a = − . < a CH [Eq. (B6) ]. The respective critical wavenumbers k ± [Eq. (B8) ] are indicated by verticalsolid lines. Panel (b) shows the stability borders a + ( k ) [Eq. (B5) ] for three different coupling strengths ρ = 0 , . and . . For a computational domain size of (cid:96) = 1 the selected wavenumbers are k n = 2 nπ .They are indicated by filled black circles in (a) and by vertical dotted lines in (b). The remaining parametersare a ∆ = − . , κ = 3 . , ¯ φ = 0 . , ¯ φ = 0 . and L = 4 π . Preprint– contact: t [email protected] – October 5, 2020 n the variational case ( α = 0 in Eqs. (4)) the free energy is a Lyapunov functional, the discrim-inant in Eqs. (12) is always positive, all eigenvalues are real and instabilities are always stationaryas for all gradient dynamics systems. A typical dispersion relation where both eigenvalues showbands of unstable wavenumbers is given in Fig. 2 (a).Stability borders a + ( k ) [see Eq. (B5)] for various variational coupling strengths ρ are given inFig. 2 (b). They always show a single maximum at zero wavenumber, i.e. the critical wavenumberis k c = 0 . This shows that the variationally coupled system only exhibits stationary large-scale(CH-)instabilities as already concluded in the previous section. One notes that an increase in thecoupling strength moves the instability onset to higher temperatures a CH [Eq. (B6)] and broadensthe band of unstable wavenumbers, i.e., the coupling acts destabilizing.The sign of ρ does not influence the range and strength of instability, however, it influencesthe character of the resulting structures as it determines whether in-phase ( ρ > ) or anti-phase( ρ < ) modulations of the two fields are favored. Overall, in the case of passive coupling thelarge-scale instability of the one-field CH equation also characterizes the two-field case. Then itcan be expected that coarsening prevails in the nonlinear evolution at large times. B. Active system
Next, we consider the nonvariational case, i.e., α (cid:54) = 0 . Then, no Lyapunov functional exists,i.e., no energy minimization guides the dynamics. As a result, oscillatory behavior can occur,as indicated by complex eigenvalues. We will also see, that furthermore one encounters a linearsuppression of coarsening.As discussed in Sec. II the linear behavior for weak nonvariational coupling | α | < | ρ | is quali-tatively equal to the stationary large-scale instability of the variational case [Fig. 2].The emergence of the maximum at finite k = k c (cid:54) = 0 [cf. Eq. (17)] in the stability border a + ( k ) marks the transition from a large-scale to a small-scale instability as illustrated by Fig. 3 (b). For α = 1 . (red line) the linear behavior is a large-scale instability. Increasing the nonvariationalcoupling to α = 1 . (green line) one observes a wide k -range where the stability border is nearlyhorizontal marking the transition to the small-scale instability. A maximum at k c (cid:54) = 0 is fullyformed for α = 1 . (blue line). Fig. 3 (a) presents a corresponding dispersion relation for a = 1 . .There, only a band of wavenumbers bound away from k = 0 shows positive growth rates. It islikely that the transition is accompanied by a suppression of coarsening which we term linear Preprint– contact: t [email protected] – October 5, 2020 k + k − k − . . . . λ n =1 n =2 n =3 (a) n =1 n =2 n =3 k . . . . . . . . a s t a b l e unstable(b) α =1.4 α =1.5 α =1.6 FIG. 3:
Linear stability behavior of homogeneous states for two actively coupled Cahn-Hilliard equations,i.e., in the nonvariational case ( α (cid:54) = 0 ). Panel (a) shows the dispersion relations λ ± ( k ) [Eq. (B1) ] atnonvariational coupling strength α = 1 . beyond the onset of the linear small-scale instability, i.e., herefor a = 1 . < a T [Eq. (B7) ]. The respective critical wavenumbers k ± [Eq. (B8) ] are indicated byvertical solid lines. Panel (b) shows stability borders a + ( k ) [Eq. (B5) ] for three different coupling strengths α = 1 . , . and . . Parameters are ρ = 1 . , a ∆ = − . , κ = 0 . , ¯ φ = 0 , ¯ φ = 0 and L = 4 π . Theremaining symbols and lines are as in Fig. 2. suppression of coarsening . In Sec. IV A we investigate this transition in the fully nonlinear regimeand discuss its impact on the dynamic behavior and the resulting steady states.Beside the described transition from a large- to a small-scale instability, the nonvariationalcoupling can also cause oscillatory behavior if | α | > | ρ | . Figure 4 shows two qualitatively differentcases: Panels (a) and (b) give a dispersion relation and stability boundaries, respectively, whencomplex eigenvalues appear in a band starting at zero wavenumber. In particular, panel (b) shows15 Preprint– contact: t [email protected] – October 5, 2020 k o k + k − k − . . . R e ( λ ) n =1 n =2 n =3 (a) Im( λ ) = 0Im( λ ) = 0 n =1 n =2 n =3 k − . − . − . − . − . . . . a s t a b l e unstable (b) α =1.3 α =1.439 α =1.5 k + k − k − . . . . . R e ( λ ) n =1 n =2 n =3 (c) Im( λ ) = 0Im( λ ) = 0 n =1 n =2 n =3 k − . − . − . . . . a s t a b l e un s t a b l e (d) α =1.35 α =1.4 α =1.44 FIG. 4:
The occurrence of oscillatory linear instability modes for two actively coupled Cahn-Hilliardequations ( α (cid:54) = 0 ). Panels (a) and (c) give dispersion relations [Eq. (B1) ] and panels (b) and (d)the respective associated stability boundaries [Eqs. (B5) , (B3) ] for two qualitatively different cases. In(a,b) complex eigenvalues occur in a band starting at k = 0 while in (c,d) they occur in a wavenum-ber band away from zero [cf. Eq. (B10) ]. Only the real part of the eigenvalues are shown, indicatingcomplex (real) eigenvalues by dashed (solid) lines. Panel (b) illustrates the transition from a stationarylarge-scale instability ( α = 1 . ) via a stationary small-scale instability ( α = 1 . ) to an oscillatory(Hopf) large-scale instability ( α = 1 . ). The dispersion relation in panel (a) corresponds to α = 1 . at a = − . < a T [Eq. (B7) ] < a H [Eq. (B4) ]. The remaining parameters for panels (a) and (b) are ρ = 1 . , a ∆ = 1 , κ = 3 . , ¯ φ = 0 , ¯ φ = 0 and L = 8 π . In panel (d) a band of complex eigen-values appears for α = ρ = 1 . (red line) at k (cid:39) k and widens with increasing α until its left limitreaches k = 0 at α = 1 . (blue line). The dispersion relation in panel (c) corresponds to α = 1 . at a = − . < a CH [Eq. (B6) ]. The remaining parameters for panels (c) and (d) are ρ = 1 . , a ∆ = − , κ = 3 . , ¯ φ = 0 , ¯ φ = 0 and L = 4 π . For both dispersion relations the respective critical wavenumbers k ± [Eq. (B8) ] of real and k o [Eq. (B9) ] of complex roots are indicated by vertical solid lines. The remainingsymbols and lines as as in Fig. 2. how with increasing nonvariational coupling, first, a transition occurs from a stationary large-scaleinstability ( α = 1 . ) as in Fig. 2 to a stationary small-scale instability ( α = 1 . ) as in Fig. 3.Then, a further increase in α results in the appearance of a band of oscillatory modes at k = 0 thatextends towards larger k and always represents a large-scale instability ( α = 1 . ). Depending onthe specific value of a the large-scale oscillatory instability or the small-scale stationary instability16 Preprint– contact: t [email protected] – October 5, 2020 an be dominant, i.e., have the larger maximal growth rate.The dispersion relation for α = 1 . and a = − . in panel (a) illustrates the latter case with dominant small-scale instability. Note thatthe pure small-scale instability at intermediate α in Fig. 4 (b) is not always part of the transitionscenario from stationary to oscillatory large-scale instability.Panels (c) and (d) of Fig. 4 illustrate the second way how oscillatory modes can appear, namely,in a wavenumber band bound away from k = 0 . In panel (d), the red line for α = ρ = 1 . showsthe stability border when all modes are still real and the large-scale instability occurs. As soon as α > ρ , e.g., at α = 1 . (green line), a band of oscillatory modes occurs. Since the maximum ofthe stability boundary remains at k = 0 and the eigenvalues at small wavenumbers remain real, atonset (at a ≈ . ) one still has a large-scale stationary instability. If we consider the dispersionrelation in panel (c) for α = 1 . and a = − . (far above onset), we see that although the globalmaximum of the growth rate corresponds to a stationary mode, the band of oscillatory modesbegins nearby and contains another maximum. Therefore, it can be expected that time-periodicbehavior becomes important. Furthermore, the band of complex eigenvalues with positive realparts causes both real eigenvalues λ ± ( k ) at small k to be positive.Further increasing α , the band of complex eigenvalues widens. Its lower border reaches k = 0 when φ − φ − a ∆ < √ ∆ and the large-scale stationary instability becomes an oscillatory one.The impact of complex eigenvalues and the onset of time-periodic behavior in the fully nonlinearregime is discussed in Sec. IV C. III. NONLINEAR BEHAVIOR IN VARIATIONAL CASE
Next, we briefly review the phase behavior of the coupled CH model in the variational case( α = 0 ). In particular, we discuss the phase behavior in the thermodynamic limit via a study ofthe coexistence of uniform states, i.e., for an infinite domain where interfaces can be neglected.Furthermore, the phase behavior is related to the bifurcation behavior found for finite systems.To calculate uniform steady states in the thermodynamic limit, we set all spatial derivatives inEqs. (7) to zero and obtain f (cid:48) ( φ ) − ρφ − µ , f (cid:48) ( φ ) − ρφ − µ .. (23)Next, we consider two uniform states in different boxes “A” and “B” with concentrations φ A , φ A and φ B , φ B , respectively. At coexistence, the two boxes are at equal temperature (by definition for17 Preprint– contact: t [email protected] – October 5, 2020 ur isothermal system), at identical chemical potentials µ and µ and at identical pressure p , i.e.,equal grand potential density ω = − p = f + f − ρφ φ − µ φ − µ φ . (24)As a result we have the three conditions µ A1 = µ B1 ,µ A2 = µ B2 , (25) p A = p B to determine the four unknown concentrations at coexistence leaving one of them a free parameter.The resulting phase diagrams represented in planes spanned by the mean concentrations and thechemical potentials are given in Figs. 5 (a) and (b), respectively. Further details on the continuationprocedure to obtain the phase boundaries see Ref. [55].For the specific choice a ∆ = 0 the steady equations (23) are symmetric w.r.t. an exchange ofthe two fields ( φ , φ ) → ( φ , φ ) additionally to the symmetry w.r.t. inversion of Eqs. (4). Thesesymmetries are inherited by the phase diagrams. Namely, they are reflection symmetric w.r.t. bothdiagonals. In the four corners of panel (a) one finds the four phases I to IV with various extendedcoexistence regions in between. The four phases may be called (I) high- φ , high- φ phase, (II)low- φ , high- φ phase, (III) low- φ , low- φ phase, and (IV) high- φ , low- φ phase.For the present ρ > case, all phases with the exception of II and IV can pairwise coexist (foran ρ < the excluded combination will be I-III). This is best seen in Fig. 5 (b) where heavy solidlines indicate phase boundaries in the ( µ , µ ) -plane where two phases coexist and green trianglesymbols triple points where three phases coexist. In the ( ¯ φ , ¯ φ ) -plane [Fig. 5 (a)] two coexistingstates lie on binodal lines (heavy solid lines ) and are connected by tie lines (thin dashed lines) thatrepresent the Maxwell construction in the ternary system. States between binodals are unstablew.r.t. phase decomposition and would decompose along the tie lines. The triple points in Fig. 5 (b)become extended regions in the representation of Fig. 5 (a) (green shaded area). Correspondingstates decompose into the three coexisting states at the corners of the triangle.It is interesting to note that for large | φ | or | φ | the two fields practically decouple. For in-stance, for φ (cid:29) , to leading order φ is uniform, and φ separates into states 1 and -1. This isalready well visible in Fig. 5, even at ¯ φ = 2 . and can also be seen in the concentration profilesdiscussed below. Actually, in the slightly artificial limit ρ → the two fields entirely decouple18 Preprint– contact: t [email protected] – October 5, 2020 − ¯ φ − − ¯ φ IIIIII IV(a) − − µ − − − − µ III III IV(b) − − ¯ φ . . . . . || δ φ || (c) − . . . − φ i φ φ − . . . − − . . . − φ i − . . . − − . . . x − φ i − . . . x − (d) FIG. 5:
The phase behavior of the ternary system modeled by two variationally coupled Cahn-Hilliardequations [Eq. (23)] represented in planes spanned by (a) the mean concentrations and (b) the chemicalpotentials. Phases I to IV are described in the main text. The heavy solid lines represent the various (a)binodals and (b) the thermodynamic phase boundary, i.e., they represent coexisting stable states. The thindotted lines are coexisting unstable states while the straight dashed lines in (a) are tie lines connectingparticular coexisting states. The triangular green shaded regions in (a) indicate three-phase coexistenceand corresponds to the triple points (green triangle symbols) in (b). The remaining parameters are a = − , a ∆ = 0 , and ρ = 1 . . (c) Bifurcation diagram with control parameter ¯ φ at fixed ¯ φ = 0 [i.e., straightvertical cut of (a)] and finite domain size L = 10 π (and κ = 1 ). Panel (d) gives examples of concentrationprofiles at points marked by plus symbols in (c). The remaining parameters are as in (a) and (b). Preprint– contact: t [email protected] – October 5, 2020 nd the ( φ , φ ) -phase diagram converges to vertical and horizontal binodal lines at ±√− a . Theircrossing points define a square that contains a “four-phase coexistence” region. In the ( µ , µ ) -plane vertical and horizontal lines at zero chemical potential cross at the origin that correspondsto a quadruple point of four-phase coexistence.Now we come back to the case of ρ > in Fig. 5 to discuss the remaining features: Theconcepts of binodals and coexistence can be extended beyond the thermodynamic limit: First,two phases may still coexist even if the resulting state does not correspond to the global energyminimum anymore, because either coexistence involves another phase or the homogeneous mixedstate becomes the global minimum. Corresponding metastable coexistence states are given as thinsolid lines in Figs. 5 (a) and (b). They exist for a small parameter range after the binodals cross atriple point. If at least one of the “coexisting” states is linearly unstable, coexistence is drawn asdotted line. Each “unstable binodal” connects the ends of two “metastable binodals” and representsa threshold state that has to be overcome when going from a metastable coexistence to a stableone. Knowledge of such metastable and linearly unstable states is particularly important when thedynamics of phase transitions is considered, e.g., when considering the motion of fronts. In thefully decoupled limit, metastable coexistence is limited by an “inner square” at | φ i | = (cid:112) − a/ within the 4-phase coexistence region.Second, one can consider phase coexistence in finite systems, where interface energies relatedto the transitions between regions of different phases become important, i.e., the gradient-squaredterms in the energy functional. Then, the transition between phases can be described by a bifur-cation diagram giving a property of states over a control parameter like the one in Fig. 5 (c) withexample profiles given in panel (d). Increasing the system size, one can systematically study thetransition from such bifurcation diagrams obtained at finite system sizes to the thermodynamiclimit. See Ref. [17] for examples including phase separation in a binary system, i.e., a one-fieldCH equation.Here, the bifurcation diagrams provide us with the passive reference case for the active systeminvestigated in section IV. In Fig. 5 (c) we use the mean concentration ¯ φ as control parameter,keep ¯ φ = 0 fixed, and employ a suitable norm as solution measure [see Eq. (8)]. With otherwords, we consider a vertical straight cut through the phase diagram in Fig. 5 (a). At small ¯ φ , westart in the coexistence region of phases III and IV, i.e., the low- φ , low- φ phase and the high- φ , low- φ phase coexist as exemplified in the upper left panel of Fig. 5 (d). There, the φ ( x ) and φ ( x ) profiles are given as blue dotted and orange solid lines, respectively. Close inspection20 Preprint– contact: t [email protected] – October 5, 2020 . − . − . − . . . a . . . . . || δ φ || n =148101215 (a) − . − . . . . x − . − . . . . . φ i (b) φ φ − . − . . . . x − . − . − . . . . . φ i (c) − . − . . . . x − − φ i (d) FIG. 6:
Panel (a) shows a bifurcation diagrams of steady states for the ternary system in the variationalcase employing the effective temperature a as control parameter at fixed L = 30 π , κ = 1 , ¯ φ = 1 , ¯ φ = − . , a ∆ = − . , and ρ = 1 . Solid (dashed) lines indicate linearly stable (unstable) states. Theblack horizontal line represents the homogeneous state. The various blue lines represent phase-separatedstates with different numbers of phase-separated structures, i.e., periods ( n = 1 to n = 15 branch). Panels(b) to (d) give profiles of selected linearly stable states at loci indicated in (a) by crosses. shows that the concentration values of the plateaus are not exactly the binodal values in the phasediagram. This is due to the finite size of the system.Increasing ¯ φ , we cross the triple point region of the phase diagram, then enter the I-III coex-istence. In the bifurcation diagram, the branch undergoes two saddle-node bifurcations where thestates loose and regain linear stability, respectively. The corresponding S-shaped structure is re-lated to the nucleation of a third phase within the profile, namely, the high- φ , high- φ phase (i.e.,phase-I) that appears at the center of the phase-IV plateau [see upper right panel of Fig. 5 (d)].21 Preprint– contact: t [email protected] – October 5, 2020 hase IV is still visible as two shoulder-like plateaus between the expanding phase I and phase III.Further increasing ¯ φ , the plateaus of phase IV shrink and are replaced by phase I [center leftpanel of Fig. 5 (d)]. Beyond the maximum at ¯ φ = 0 , the two concentration fields reverse rolesand phase III is replaced by phase II in a similar sequence of events [center right till lower rightpanels of Fig. 5 (d)]. Beyond the r.h.s. pair of saddle-node bifurcations, we end in the coexistenceregion of phases I and II, i.e., the high- φ , high- φ phase and the low- φ , high- φ phase.Up to here, we have used the concentrations as main parameters and kept the effective temper-ature a fixed. As a will be an important control parameter in section IV we next study its influencein the passive case. If a decreases, the binodal lines in the phase diagram move apart and the threestate coexistence region becomes larger. Concentration profiles show steeper interfaces betweenphases rendering plateaus more pronounced. The bifurcation diagram in Fig. 6 (a) shows the normas a function of a at fixed domain size L = 30 π . As for the one-field model [17], with decreasing a the uniform state becomes unstable at about a = 0 . where the completely phase-separated,i.e., fully coarsened, state emerges in a subcritical pitchfork bifurcation. Decreasing a further, theuniform state becomes successively unstable with respect to higher order modes and correspond-ing branches emerge in a number of pitchfork bifurcations. We label the different branches bythe spatial periodicity n of the corresponding solutions. Overall, we exemplary show six of thefirst fifteen branches, n = 1 , , , , and . Note that for n > , all states are unstable. Ina time evolution they will coarsen and develop into states on the n = 1 branch. The n = 1 to n = 8 branches bifurcate subcritically, i.e., they emerge towards larger a before turning back atrespective saddle-node bifurcations.In particular, the n = 1 branch emerges with unstable profiles (nucleation thresholds, analogueto [56, 57]) and stabilizes at the saddle-node bifurcation at a ≈ . . The resulting stable statesshow a coexistence between the high- φ , high- φ phase I and the high- φ , low- φ phase IV, see,e.g., the profile in Fig. 6 (b). We note that the plateau concentrations are already relatively closeto the corresponding binodal values. An increase in domain size will result in full convergence.Further following the n = 1 branch with decreasing a , it eventually undergoes another pair ofsaddle-node bifurcations, thereby passing through unstable states ( − . < a < − . ) before sta-bilizing again. The corresponding hysteresis loop is again related to the nucleation of a third phase(here, phase III: low- φ , low- φ ) that emerges at the center of the phase IV plateau [Fig. 6 (c)].The remaining part of the branch shows well-developed three-phase coexistence of phases I, IIIand IV [Fig. 6 (d)]. Note that the unstable state existing in the hysteresis range corresponds to an22 Preprint– contact: t [email protected] – October 5, 2020 . − . t × φ
012 0 . . . . φ x . − . t × φ -101 − . − . − . . . . . φ (a) a = − . x . − . t × φ -2-1012 − . − . . . . . φ x . − . t × φ -2-1012 − . − . − . . . . φ (b) a = − . FIG. 7:
Space-time plots of selected time simulations illustrate the emergence of the n = 1 states of thebifurcation diagram Fig. 6 (a) after some coarsening. Left and right panels show fields φ ( x, t ) and φ ( x, t ) ,respectively. Row (a) shows the emergence of two-phase coexistence at a = − . while in row (b) a three-phase state emerges at a = − . . The domain size is L = 10 π and the remaining parameter are as inFig. 6. unstable threshold state that has to be overcome to switch between the two linearly stable states.Time simulations are performed to confirm that depending on the specific value of a , two- orthree-phase coexistence emerges after initial coarsening. The simulations are initialized with the23 Preprint– contact: t [email protected] – October 5, 2020 niform state with added white noise of small amplitude · − . If not stated otherwise this isdone in all simulations throughout the paper. Fig. 7 (a) shows that at a = − . [to the right of theS-shaped structure in Fig. 6 (a)] a two-phase n = 1 state develops after coarsening via a Volumemode from an n = 3 state. Fig. 7 (b) gives a similar simulation for a = − . [to the left of theS-shaped structure in Fig. 6 (a)]. Here, after some coarsening an n = 1 three-phase state emergesas the system is in the parameter region of the triple point. IV. NONLINEAR BEHAVIOR IN NONVARIATIONAL CASE
Next, we introduce the nonvariational case by increasing the coupling α from zero and investi-gate how breaking the gradient dynamics structure changes the bifurcation behavior. Based on thelinear results in section II, we focus on three phenomena: the arrest and the complete suppressionof coarsening (sections IV A and IV B), and the onset of time-periodic behavior (section IV C).Our analyses reveal that most qualitative changes w.r.t. the variational case occur for a nonva-riational coupling stronger than the variational one. Therefore, we now focus on | α | > | ρ | , i.e., ∆ > . In appendix D we consider the special case where no variational coupling is present, i.e., ρ = 0 and α (cid:54) = 0 . A. Arrest and suppression of coarsening
The linear analysis in section II B shows that nonvariational coupling can induce a small-scaleinstability uncommon for CH models. Next, we employ time simulations and bifurcation studies toinvestigate the consequences of this linear behavior in the fully nonlinear regime. It is instructiveto first consider the bifurcation behavior of steady states in the purely large- and small-scale casesin Figs. 2 and 3, respectively. In both cases, we use α > and consider parameter values whereboth eigenvalues are still real.The resulting bifurcation behavior for two values of α close to the transition from large- tosmall-scale instability is presented in Fig. 8 again using a as control parameter. Parameters arechosen as in Fig. 3, namely, panel (a) with α = 1 . corresponds to the green line in Fig. 3 (b) andpanel (b) with α = 1 . belongs to the dispersion curve in Fig. 3 (a) and the blue line in Fig. 3 (b).Branches emerging at primary bifurcations from the uniform state are named the periodicity n ofthe decomposition pattern as introduced in section III.24 Preprint– contact: t [email protected] – October 5, 2020 . . . . . a . . . . . || δ φ || n =1 n =5 (a) .
30 1 .
35 1 .
40 1 .
45 1 . a . . . . || δ φ || n =1 n =2 (b) − . − . . . . x − . − . − . . . . . φ i (c) − . − . . . . x − . − . . . . φ i (d) − . − . . . . x − . − . . . . φ i (e) φ φ FIG. 8:
Bifurcation diagrams for the nonvariationally coupled Cahn-Hilliard model [Eq. (4) ] as a functionof the parameter a . Linear and nonlinear suppression and arrest of coarsening are illustrated using nonva-riational coupling (a) α = 1 . and (b) α = 1 . stronger than the variational coupling ρ = 1 . . In (b) onlythe fully phase-separated state ( n = 1 , blue line) and the two-period state ( n = 2 , green line) are shownwhile in (a) states up to n = 5 are included. Primary and secondary pitchfork bifurcations are markedby circle symbols. Selected side branches are also included. The remaining line styles and parameters areas in Fig.3. The lower panels show profiles of (c) stable and (d,e) unstable steady states at loci marked inpanel (a) by crosses. As expected based on the linear result, when decreasing a in panel (a) the n = 1 state bifurcatesfirst, corresponding to a large-scale instability. The bifurcation is a supercritical pitchfork as allother considered primary bifurcations. In consequence, the shown n = 2 (green line) to n = 5 (purple line) states inherit two to eight unstable eigenvalues from the uniform state. In qualitativecontrast to the variational case, where all n > states remain always unstable, here, they even-tually stabilize at secondary pitchfork bifurcations. No such bifurcations occur in the variationalcase. In the weakly nonvariational case which we define as | α | < | ρ | we do observe secondarybifurcations (not shown). They always occur in pairs of one destabilizing and one stabilizing bi-25 Preprint– contact: t [email protected] – October 5, 2020 urcation related to higher order modes and do not result in the appearance of further stable statesas observed for | α | > | ρ | .In particular, the n = 2 state [cf. Fig. 8 (c)] stabilizes through a degenerate pitchfork bifurca-tion where two real eigenvalues cross zero and two distinct subcritical branches (brown and graylines) simultaneously emerge towards smaller values of a . Note that on the scale of Fig. 8 (a) thetwo curves can not be distinguished by eye. Also, each branch corresponds to four states relatedby symmetry (see below). Example profiles on the two secondary branches are given in Fig. 8 (d)and (e), respectively. Both states break the discrete translational symmetry of the primary n = 2 branch. The bifurcation structure can be understood considering reflection symmetries: States onthe n = 2 primary branch have two independent reflection symmetries, one with respect to theirminima and another one with respect to their maxima. For nonzero mean concentrations, two dis-tinct pitchfork bifurcations correspond to the respective breaking of these symmetries (not shown).In Fig. 8 (a), ¯ φ = ¯ φ = 0 ensures inversion symmetry and the two reflection symmetries can beidentified via an inversion. That is, they are always broken together in a degenerate pitchfork (alsotermed Z × Z bifurcation [58]) with normal form ˙ x = µx − b x − b x x ˙ x = µx − b x − b x x . Here x and x refer to the two modes of symmetry breaking, e.g. x [ x ] breaks the reflectionsymmetry w.r.t. the minima [maxima]. The primary n = 2 branch is represented by ( x , x ) =(0 , , see example profile in Fig. 8 (c). Then, there are two pairs of branches which keep eitherthe reflection symmetry w.r.t. the minima or w.r.t. to the maxima with representations (0 , ± (cid:113) µb ) and ( ± (cid:113) µb , . One of these pairs corresponds to the profile in Fig. 8 (d) and the other oneto its inversion. Furthermore, there are four branches which break both reflection symmetries,however keep full inversion symmetry, i.e., ( x, φ i ) → ( − x, − φ i ) , see example profile in Fig. 8 (e).Their representations are ( ± (cid:113) µb + b , ± (cid:113) µb + b ) . In total there are eight simultaneously emergingsecondary branches, i.e., each of the two distinct secondary branches in Fig. 8 (a) is four-fold andcan be “unfolded” choosing adequate parameters and model amendments.Comparing Figs. 8 (a) at α = 1 . and (b) at α = 1 . we see that with the increase of α thefirst two primary bifurcations have swapped position reflecting the transition from large-scale tosmall-scale linear instability [cf. Section II]. In consequence, at the first primary bifurcation the(now linearly stable) n = 2 state emerges supercritically. The fully phase-separated ( n = 1 ) state26 Preprint– contact: t [email protected] – October 5, 2020 nly emerges at the second primary bifurcation, supercritical but twice unstable. Thus, the linearsuppression of coarsening results in an extended parameter region where only the patterned n = 2 state is stable.It is noteworthy that when the primary bifurcations switch places, the above discussed sec-ondary bifurcations move from the n = 2 branch onto the n = 1 branch [Fig. 8 (b)]. In conse-quence, the two primary and two secondary bifurcations all coincide at the crossover. Four param-eters, α , a and both mean concentrations ¯ φ , ¯ φ need to be adjusted to pinpoint the correspondingcodimension-4 bifurcation point. However, when the two secondary pitchfork bifurcations haveswitched onto the n = 1 branch, they do not coincide anymore. The reason is that one cannot anymore independently break the reflection symmetries with respect to the minimum and themaximum. As a result, breaking the reflection symmetry and breaking the full inversion symmetryoccurs independently and hence, the degeneration of the secondary bifurcations is lifted. The first[second] pitchfork bifurcation breaks reflection [full inversion] symmetry and pairs of brancheswith solutions similar to Fig. 8 (e) [Fig. 8 (d)] emerge. For any nonzero mean concentration theinversion symmetry is broken for all states. Hence, in that case the second pitchfork bifurcationunfolds into a saddle-node bifurcation and a continuous branch.Coming back to Fig. 8 (a), we note that a consequence of the degenerate pitchfork bifurca-tion is the simultaneous stabilization of both coarsening modes (volume transfer and translation)mentioned in the introduction and discussed at the end of section III. Similar stabilizations areobserved for the branches of larger n where, however, involving a sequence of several subsequentbifurcations. Namely, two, three and four degenerate pitchfork bifurcations on the n = 3 , and branch, respectively, ensure that for a (cid:46) − . all n ≤ branches are linearly stable. It is in-triguing that the simultaneous stabilization of translation and volume coarsening modes is genericin a wide range of parameters. Again this is a consequence of the choice ¯ φ = ¯ φ = 0 responsiblefor the field inversion symmetry of Eqs. (4). Note that none of the studied emerging secondarybranches reconnects to the primary branch.Next we analyze further implications of the nonlinear results. They indicate that the describedlinear suppression of coarsening is only valid for a finite range of a , namely, until the n = 1 branchstabilizes via the two discussed secondary pitchfork bifurcations at a ≈ . and a ≈ . .Before this occurs, the n = 1 branch is unstable to two modes resulting in splitting of the fullyphase-separated state. One may call them “reverse coarsening” modes in analogy to the “reverseOstwald ripening” in Ref. [27]. At lower a (cid:46) . , multistability with higher- n states arises as27 Preprint– contact: t [email protected] – October 5, 2020 a) Linear suppression x . − . t × φ -0.500.5 − . . . . φ (b) Nonlinear arrest x . − . t × . . φ -101 − . . . . φ (c) Nonlinear suppression x . − . t × . . φ -101 − . . . . φ FIG. 9:
Space-time plots obtained by direct numerical simulation of structuring processes in the nonvari-ational case. They illustrate three qualitatively different behaviors that replace the classical coarsening ofthe variational case: (a) linear suppression of coarsening at α = 1 . and a = 1 . [cf. Fig. 8 (b)], (b)nonlinear arrest of coarsening at α = 1 . and a = 1 . [cf. Fig. 8 (a)], and (c) nonlinear suppression ofcoarsening at α = 1 . and a = 1 . [cf. Fig. 8 (a)]. For details see main text. before.Fig. 9 illustrates consequences of multistability for the time evolution of active phase separa-tion, in particular, the coarsening behavior. Panel (a) shows a corresponding space-time plot at a = 1 . and α = 1 . , a region in Fig. 8 (b) where n = 1 and n = 2 state both exist, but onlythe patterned n = 2 state is stable. The chosen a lies between the two secondary bifurcations, i.e.,the n = 1 state has one unstable eigenvalue. Starting with the n = 1 state with added noise, weobserve reverse coarsening via the mass transfer mode converging to the patterned n = 2 state.This clearly illustrates that the nonvariational coupling can reverse the original coarsening processof a phase separating system. It is a direct result of the linear suppression of coarsening discussedabove, because the stability of the relevant branches is a direct consequence of the linear stabilityof the uniform state.Next, we consider a time evolution in the multistable region of Fig. 8 (a). Figs. 9 (b) and (c)present results for a = 1 . and a = 1 . , respectively. In both cases, first an n = 3 state developscorresponding to the fastest growing linear mode. However, as at a = 1 . the n = 3 state isstill unstable, in Fig. 9 (b) a single coarsening step occurs towards the linearly stable n = 2 statewhere coarsening is arrested. We term this behavior nonlinear arrest of coarsening . In contrast, Note that at the parameters of Fig. 9 (b) one may also start with a large-amplitude n = 1 mode. Then the systemevolves into the linearly stable n = 1 state (not shown) as expected in a multistable region. Preprint– contact: t [email protected] – October 5, 2020 t a = 1 . [Fig. 9 (c)] after an initial transient the now linearly stable n = 3 state forms andno coarsening occurs. This case we term nonlinear suppression of coarsening as it depends onthe sequence of secondary bifurcations and can not be discerned from solely analyzing the linearbehavior of the uniform state.To summarize, the bifurcation diagram and simulation results show that an arrest or suppressionof coarsening can occur even if the dispersion relation for the uniform state indicates a large-scaleinstability and one would naturally predict coarsening. The underlying mechanism is nonlinearand can be characterized as follows. In the common coarsening process, clusters of the same phasemerge over time, and their number successively decreases until the fully phase-separated state isreached. This implies that the eigenvalues of all coarsening modes become very small for stateswith a small number of clusters, but they always remain positive. Here, the nonvariational couplingdisrupts the coarsening before the n = 1 state is reached because all relevant eigenvalues havebecome negative. Thus, the onset of multistability marks the arrest or suppression of coarseningdepending on the fastest growing linear mode.The same mechanism also acts for large- n states. In Fig. 8 (a) we observe it up to the n = 5 branch where the fourth degenerate secondary pitchfork bifurcation marks the arrest of coarseningat the state with five peaks. Note that below in section IV C we discuss more intricate, time-periodic behavior. Then our simple explanation how coarsening is suppressed is not valid anymore.However, next we focus on another interesting system property related to the behavior of theprimary bifurcations. Up to here, all considerations of suppression of coarsening have focused onsituations where all primary bifurcations are supercritical. For phase separation phenomena thisis often not the case. Therefore, we next analyze whether one observes a similar suppression ofcoarsening if primary bifurcations are subcritical. Furthermore, we show that subcritical primarybifurcations may occur without quadratic nonlinearity, i.e., here at mean concentrations ¯ φ = ¯ φ =0 . This qualitatively differs from behavior known for the classical one-field CH equation. B. The subcritical case
In the classical passive one-field CH equation, subcritical primary bifurcations occur for meanconcentrations | φ | > / √ [56] (for details, use D = 0 in the derivation in the appendix of [21]).In general, it is known that quadratic nonlinearities (in general, nonlinearities of even power) breakthe field inversion symmetry and lead to subcritical behavior [54]. In the CH case, moving at least29 Preprint– contact: t [email protected] – October 5, 2020 . . . . . . . a . . . . . . . . || δ φ || (a) n =1 n =2 . . . . . . a . . . . . . . . || δ φ || (b) n =1 S n =2 − . . . . . . a . . . . . . . . || δ φ || (c) n =1 S n =2 S n =3 n =4 − . − . − . . . . a . . . . . || δ φ || (d) n =1 + n =1 − n =2 + n =2 − n =3 + n =3 − FIG. 10:
Subcritical bifurcation behavior of steady states for the nonvariationally coupled CH model.Panels (a) and (b) show for ρ = 1 . cases of a large- and a small-scale linear instability at α = 1 . and α = 1 . , respectively. Mean concentrations are ¯ φ = 0 and ¯ φ = 0 . (cid:54) = 0 , with remaining parameters asin Fig. 8. Panel (c) gives more intricate behavior at ρ = 1 . and α = 1 . , parameters otherwise as (a,b).Relevant branches are labeled by their periodicity n and a subscript “S” if they emerge in a secondarypitchfork bifurcation. Panel (d) shows that subcritical behavior may for κ = 1 even arise at ¯ φ = ¯ φ = 0 ;other parameters are as in (a,b). Circles, triangles and diamonds indicate pitchfork, drift pitchfork andHopf bifurcations, respectively. one mean concentration away from zero indeed breaks the field inversion symmetry and thereforefacilitates the occurrence of subcritical bifurcations. This can be clearly seen when transformingEqs. (4) using shifted concentration fields such that the new homogeneous state is always at zero.The old mean concentrations then appear as parameters and the old purely cubic nonlinearitiesunfold into a cubic polynomials containing quadratic and linear terms.30 Preprint– contact: t [email protected] – October 5, 2020 f the quadratic term passes a certain threshold, e.g., for a range of nonzero ¯ φ , primary bifur-cations can become subcritical. Here, we chose ¯ φ = 0 . and accordingly adapt α to investigatethe transition from large- to small-scale instability. The linear behavior is similar to the case dis-cussed at Fig. 3 in Section II B. Bifurcation diagrams characterizing the nonlinear behavior nearthe transition are shown in Fig. 10. Panel (a) and (b) give results for α = 1 . and α = 1 . ,respectively, showing all branches that eventually connect to the homogeneous state at the first orsecond primary bifurcation. Between the two panels a transition occurs analog to the one betweenFigs. 8 (a) and (b) for the supercritical case.In Fig. 10 (a) the n = 1 branch (blue line) bifurcates first and coarsening can proceed un-hindered as all other states are unstable in a large a -range (large-scale instability). The branchbifurcates subcritically and gains stability at a saddle-node bifurcation at about a ∼ . . Atthe second primary instability the n = 2 state (green line) emerges subcritically with unstableeigenvalues, which are stabilized through two secondary pitchfork bifurcations and a saddle-nodebifurcation, finally resulting in linear stability for a (cid:46) . . At the two well-separated secondarybifurcations the n = 2 state is stabilized with respect to the two coarsening modes. The secondarybranch which emerges in Fig. 10 (a) at the first secondary bifurcation very close to the secondprimary bifurcation [see inset] emerges due to the stabilization of the volume mode of the primarybranch.In contrast, Fig. 10 (b) at α = 1 . illustrates a case beyond the transition where the linearanalysis of the uniform state indicates occurrence of a small-scale instability. Although, overallthe appearance and stability are rather similar to Fig. 10 (a), inspection of the inset shows that thelocal bifurcation behavior has strongly changed: At the first primary bifurcation, now the n = 2 state subcritically emerges carrying one unstable eigenvalue. Shortly after, a secondary supercrit-ical pitchfork bifurcation occurs, where the blue n = 1 S branch supercritically emerges inheritingthe one unstable eigenvalue. Comparing to panel (a), we still consider it as the the fully phase-separated state n = 1 but indicate by the subscript “S” the qualitative different emergence in asecondary instead of a primary bifurcation. Nevertheless, as before, the n = 1 S branch fully stabi-lizes at the saddle-node bifurcation and in a wide a -range it is the only stable state. In the secondprimary bifurcation, the n = 1 branch (brown line) emerges supercritically carrying two and, aftera nearby saddle-node bifurcation, three unstable eigenvalues, i.e., it has similar properties as inFig. 10 (a) where it emerges at the first secondary bifurcation of the n = 2 state. One may saythat at the transition from large- to small-scale instability the primary n = 1 bifurcation and the31 Preprint– contact: t [email protected] – October 5, 2020 rst secondary bifurcation on the n = 2 branch exchange their roles. Only two parameters ( α and a ) are adjusted to reach the transition point that again displays properties of a higher codimensionpoint as three bifurcations coincide, namely two primary ones and a secondary one. This is due tothe additional special condition provided by ¯ φ = 0 .We see that the merely local changes at the transition are largely overshadowed by mainlyundisturbed global behavior related to the subcriticality. Hence, due to the branches which emergefrom secondary bifurcations no linear suppression of coarsening occurs. Only if the primary bifur-cations are supercritical, the switch in linear behavior from large- to small-scale instability directlyresults in the linear suppression of coarsening. The nonlinear effects of suppression and arrest ofcoarsening already discussed in the supercritical case are unaffected by the subcriticality. For in-stance, the final secondary bifurcation of the n = 2 branch where it becomes linearly stable stillmarks the onset of nonlinear arrest or suppression of coarsening.Fig. 10 (c) evidences that reordering of the primary bifurcations can be more extensive. In-creasing ρ and α as compared to Figs. 10 (a) and (b), now at the first primary bifurcation the n = 3 branch emerges subcritically carrying a secondary bifurcation where the n = 1 S branchemerges as well as another branch that connects to the first secondary branch of the second pri-mary branch. The latter is actually the n = 4 branch on which the n = 2 S branch emerges.However, when crossing through the stable parts at large norm they are still well ordered: fromright to left n = 1 S , S , , . . . .It is remarkable, that in the present nonvariationally coupled system subcritical behavior caneven occur at zero mean concentrations, i.e., where the above argument regarding the quadraticnonlinearity does not hold. This is illustrated in Fig. 10 (d) and mathematically illuminated byweakly nonlinear analysis in appendix C. The derived amplitude equations [see Eq. (C11)] defineparameter ranges illustrated in Fig. 11 where this unexpected behavior occurs. At the core of theargument is a projection that is performed when applying the Fredholm alternative. If the necessarycriterion ∆ > is fulfilled this projection can produce nonlinearities that act destabilizing toleading order and result in subcritical behavior (even if the nonlinearities in the original equationsappear stabilizing). Such projections can only occur if the model couples at least two fields.The bifurcation diagram in Fig. 10 (d) for κ = 1 and ¯ φ = ¯ φ = 0 shows six primary bi-furcations, three being subcritical. The various branches are marked by their periodicity n and asuperscript “ + ” or “ − ” that indicates which eigenvalue [ λ + or λ − in Eq. (B1)] crosses zero at thecorresponding primary bifurcation. In previous diagrams the distinction was not needed since all32 Preprint– contact: t [email protected] – October 5, 2020 − branches emerged far away from the instability onset and were not further considered. Here,however, for each n a supercritical n − and a subcritical n + branch emerge close to each other.Since λ + > λ − for all real eigenvalues, the first bifurcation of each pair is always the n + state.Since the necessary conditions for subcritical behavior and for primary Hopf bifurcations areidentical ∆ > (see Sec. II) it is not surprising that pairs of structured states emerge close to-gether. To create a primary Hopf bifurcation two pitchfork bifurcations belonging to the same n (i.e., n + and n − ) have to collide. For all ρ (cid:54) = 0 one of these two branches displays subcritical be-havior right before collision. Note that Fig. 10 (d) shows the particular case κ = 1 . Then the onsetof subcritical behavior as well as the creation of the primary Hopf bifurcations is independent of n [cf. discussion in appendix C]. Although all primary bifurcations are still stationary we alreadyobserve time-periodic behavior at secondary and tertiary bifurcations. The inset shows two sec-ondary bifurcations on the n = 1 − branch which are connected to one degenerated pitchfork bifur-cation on the n = 2 + branch. Again the degeneracy is caused by additional symmetries resultingfrom zero mean concentrations [cf. discussion in Sec. IV A]. On both connecting branches (browndashed lines) Hopf bifurcations marked by filled diamonds occur. Similar bifurcation structuresare found on all branches which connect an n − branch with an ( n + 1) + branch (see e.g. con-necting branches between n = 2 − and n = 3 + branch). Furthermore, on the n = 1 − branch adrift pitchfork bifurcation marked by a triangle occurs. The emerging branch (gray dashed line)represents stationary drifting states. All of these time-dependent states are unstable, at least in thevicinity of their emergence. Summarized, Fig. 10 (d) implies that time-periodic behavior can arisein various ways when the nonvariational coupling strength is increased. This is further investigatedin Sec. IV C.Fig. 11 considers the parameter plane spanned by ∆ = α − ρ and Σ = ( ρ + α ) toidentify where subcriticality occurs. The orange [blue] shaded region indicates where the n − -[ n + ]-branch shows subcritical behavior. Both regions are limited at high ∆ by the horizontalHopf-threshold [Eq. (B2)]. The shape of these regions only depends on the composed parameter M = k n L (1 − κ ) − a ∆ . For the special case κ = 1 presented in Fig. 11 simply M = − a ∆ , i.e., it isindependent of the periodicity of the linear mode. Then, for a ∆ < all n + branches in Fig. 10 (d)emerge subcritically. In the case of general f and f the relevant parameter is M = k n L (1 − Qκ ) + f (cid:48)(cid:48) − Qf (cid:48)(cid:48) valid in similar ways forthe present coupled CH equations, as for coupled Swift-Hohenberg or conserved Swift-Hohenberg equations. Preprint– contact: t [email protected] – October 5, 2020
Σ = ( ρ + α ) . . . . . ∆ = α − ρ Hopf-instability ρ = 0 . ρ = 0 FIG. 11:
Illustration of criteria for subcriticality [cf. Eq. (C19) ] and Hopf-instability [cf. Eq. (B2) ] in theplane spanned by ∆ and Σ at ¯ φ = ¯ φ = 0 . Shown is the special case of equal rigidities, κ = 1 , and M = − a ∆ = 1 . . Blue [orange] regions imply subcriticality of n + [ n − ] branches. The hatched regionindicates occurrence of Hopf-instability, i.e. where ∆ > a / [cf. Eq. (B2) ]. The left inset suggests thatsubcriticality can be observed even in the immediate vicinity of α ≈ − ρ . The right inset shows that thecolored regions do not overlap and only without variational coupling, i.e. for ρ = 0 (blue dotted line),subcritical behavior does not occur. Furthermore, the analysis in appendix C reveals two further remarkable features: First, at M = 0 , i.e., for identical subsystems, no subcritical regions exist. Second, for purely nonvari-ational coupling (i.e. ρ = 0 ) no subcritical behavior precedes the appearance of primary Hopfbifurcations. This is indicated by the dotted blue line in Fig. 11 which passes the Hopf-thresholdwithout crossing the shaded regions. 34 Preprint– contact: t [email protected] – October 5, 2020 − − − a || δ φ || (a) n =1 n =7 x . − . t × φ -2-1012 (b) − − φ FIG. 12: (a) Bifurcation diagram containing seven primary branches for n = 1 . . . and the trivial one(black horizontal line) for a large a -range. Solid [dashed] lines indicate stable [unstable] states. Pitchfork[Hopf] bifurcations are marked by circles [filled diamonds]. The magnification in the inset displays thebifurcation behavior of Fig. 8 (a) and, additionally, the first four Hopf bifurcations. Panel (b) presentsa space-time plot of a simulation at a = − . , i.e. between the two stable regions of the n = 5 state.It is initialized with white noise and after resting transient converges to a drifting oscillatory state. Theremaining parameters are as in Fig. 8 (a). For any fixed ρ and increasing | α | the system follows curves in Fig. 11 described by ∆(Σ) = ∓ ρ √ Σ + Σ for α ≷ − ρ (26)For instance, the dashed black lines at fixed ρ = 0 . demonstrate that either the orange [for α < − ρ ] or the blue [for α > ρ ] shaded region is crossed before passing the Hopf threshold.These pathways of onset represent two different scenarios focused on in the following section. C. Time-dependent states
After having discussed arrest and suppression of coarsening due to nonvariational coupling, wenext analyze under which conditions such coupling causes time-periodic behavior like travelingand standing waves. Again, we normally consider situations with variational and nonvariationalcoupling both present. As before, we employ numerical path continuation and direct time simula-35
Preprint– contact: t [email protected] – October 5, 2020 ion to characterize the fully nonlinear behavior. In addition to path continuation for steady statesemployed in previous sections, here, it is also applied for time-periodic states. For a descriptionof these techniques see [52].The linear considerations in Section II have shown that ∆ > is a necessary condition for aoscillatory instability of the uniform state and that oscillatory modes may occur in a wavenumberband [ k o − , k o + ] with k o − either zero or nonzero. The corresponding condition for the Hopf bifurca-tions related to the large-scale mode is f (cid:48)(cid:48) < − Qf (cid:48)(cid:48) given that ∆ > − f (cid:48)(cid:48) f (cid:48)(cid:48) [Eqs. (14) and (15)].In general, we find that also in the nonlinear regime time-periodic behavior only occurs for ∆ > .However, nonlinearly it can emerge at lower activity than in the linear regime.First, we revise the case in Fig. 8 (a) where we have found nonlinear arrest of coarsening. Weexplained that all steady n > states are stabilized by n − secondary degenerate pitchforkbifurcations. This is the complete picture for a > − . , the range presented in Fig. 8 (a). Incontrast, Fig. 12 (a) presents a much larger a -range down to a ≈ − . Shown are the branchesof homogeneous states and of structured states with n = 1 to n = 7 . We note that a number ofHopf bifurcations (marked by filled diamonds) exist on the n = 5 , and branches. This impliesthat the simplified picture of successively extended multistability and the accompanying nonlineararrest of coarsening has to be amended as time-periodic behavior occurs for structured states oflarger n . Further, there are some additional pitchfork bifurcations, e.g., on the n = 3 branch (at a ≈ − and a ≈ − ), that do not result in the successive stabilization of all coarsening modesdescribed in section IV A. This means in the strongly nonlinear region the arrest of coarsening isamended by other phenomena also caused by the nonvariational influence.The inset of Fig. 12 (a) magnifies the a -range where the first four Hopf bifurcations occur. Wediscuss, in particular, the bifurcations of the purple n = 5 branch. Starting at the primary bifur-cation where it emerges, four stabilizing degenerate pitchfork bifurcations occur that eventuallystabilize the branch in full accordance with section IV A. Then, after a small range of stability thefirst (destabilizing) Hopf bifurcation occurs. Soon after, a second Hopf bifurcation re-establishesstability. From there on the n = 5 branch remains stable.Fig. 12 (b) illustrates the time-periodic behavior found in the window of oscillatory unstable n = 5 states. Initialized with white noise of small amplitude, first, the fastest linear mode growsresulting in the development of the a steady n = 6 state (barely visible in the figure). Beingunstable, its appearance is transient and at t ≈ . × it has coarsened into the steady n = 5 state where the spatial coarsening is arrested. However, also the steady n = 5 state is linearly36 Preprint– contact: t [email protected] – October 5, 2020 nstable and therefor transient. Oscillations in the form of a standing wave are well developed at t = 2 × . However, at t ≈ × it turns out that also the standing wave is only a transient andone observes the onset of a slow drift. Finally, a drifting n = 5 state develops, that represents amodulated wave. Its motion is a superposition of an oscillation and a drift of (on average) constantspeed.Next we scrutinize the onset of such time-dependent behavior focusing on the fully phase-separated ( n = 1 ) state in the case of zero mean concentrations at parameter values where theuniform state displays a large-scale instability. More intricate behavior may be expected if theserestrictions are lifted, however, this shall not concern us here.As explained in the previous section, the onset of time-periodic behavior is for all ρ (cid:54) = 0 preceded by the occurrence of subcritical primary bifurcations. Before two primary pitchforkbifurcation collide to form a Hopf bifurcation one of them becomes subcritical. The intricatedetails of this qualitative transition are shown for the n = 1 branch in Fig. 13 and 15. Shownare sequences of bifurcation diagrams for increasing nonvariational coupling (passing ∆ = 0 )corresponding to a route via subcriticality of the n − -branch (called scenario A ) and the n + -branch(scenario B ) [cf. Fig. 11], respectively. The conditions for the scenarios to occur areScenario A : ( M ≷ and α passes ∓ ρ ) Scenario B : ( M ≷ and α passes ± ρ ) (27)We begin with scenario A shown in Fig. 13 where M < for all k ≥ k n =1 = 2 π , and α isincreased from panel (a) ( α < ρ ) to (e) ( α > ρ ). To better understand the bifurcation behavior wefirst develop an argument from the linear analysis at approximately equal coupling strengths: For ∆ ≈ the coupling term in the φ -equation approaches zero, i.e., φ decouples from φ (but not φ from φ ) and φ even at static φ . Hence, one eigenfunction has amplitudes (1 , and the linearregime within this subspace is equivalent to the one for a single-field Cahn-Hilliard equation for φ with eigenvalue λ . The other is λ , the eigenvalue of the uncoupled CH equation for φ witheigenvector is ( − ρ/λ , . That is, both eigenvalues corresponds to uncoupled CH equations, butone of the eigenvector does not decouple if ρ (cid:54) = 0 . For the present M < , then λ + = λ and λ − = λ .Fig. 13 (a) for α = 1 . < ρ = 1 . corresponds to a completely real dispersion relations (notshown). The steady n = 1 + and n = 1 − branch both emerge at supercritical pitchfork bifurcationsat a -values where λ + and λ − cross zero at k = k n =1 = 2 π (cf. Eq. (B1)), respectively. The stable37 Preprint– contact: t [email protected] – October 5, 2020 . − . − . − . − . − . − . a . . . . . . || δ φ || (a) n =1 + n =1 − − . − . − . − . − . − . − . a . . . . . . || δ φ || (b) − . − . − . − . − . − . − . a . . . . . . || δ φ || (c) − . − . . . . − . − . − . − . − . − . − . a . . . . . . || δ φ || (d) − . − . − . − . − . − . − . a . . . . . . || δ φ || (e) − . − . − . − . − . − . a . . . . . . . ε (f) FIG. 13:
Panels (a)-(e) show a sequence of bifurcation diagrams for scenario A of the emergence of sub-criticality and time-dependent behavior for increasing nonvariational coupling α = 1 . , . , . , . and . at ρ = 1 . and M < . Solid [dotted] lines represent linearly stable [unstable] states. Pitchfork,drift pitchfork, saddle-node and Hopf bifurcations are marked by circle, triangle, cross and filled diamondsymbols, respectively. The inset in (c) further marks by plus symbols states emerging in time simulations,e.g., shown in Fig. 14. The remaining parameters are L = 4 π , a ∆ = − . , ¯ φ = ¯ φ = 0 , κ = 3 . .Panel (f) displays the loci of all local secondary bifurcations in the ( α, a )-plane using the colors of thecorresponding primary branch. It indicates that all five secondary bifurcations visible in (c) emerge fromthe point of high codimension marked by the square symbol in (b). Preprint– contact: t [email protected] – October 5, 2020 . − . t × φ -101(a) − . − . . . . φ x . − . t × φ -101(b) − . − . . . . φ x . − . t × φ -101(c) − . − . . . . φ FIG. 14:
Space-time plots illustrating selected behavior emerging for various a -values for the bifurca-tion diagram of Fig. 13 (c). We find (a) at a = − . a stationary traveling wave [i.e., on solid part ofgray branch in Fig. 13 (c)]; (b) at a = − . a modulated traveling wave [i.e., on light blue branch inFig. 13 (c)]; and (c) at a = − . a standing wave [i.e., on solid part of red branch in Fig. 13 (c)]. n = 1 + branch features fully phase-separated states dominated by field φ , while the unstable n = 1 − branch consists of states where both fields have similar amplitudes. At first sight, thebehavior is qualitatively similar to phase separation in the purely variational case although α isalready quite large. Note, however, that at the chosen concentration values, the passive systemwould separate into phases I and III (not shown, cf. Fig. 5(a)). Here, this is not the case asthe nonvariational coupling effectively decouples φ from φ as discussed above. However, nooscillatory states appear.Increasing α , the two primary bifurcations slowly move towards each other, while the n = 1 + branch develops a bulge that extends towards the n = 1 − branch, that itself increases the curvatureof its leftward bend. Eventually, at α = ρ the n = 1 + bulge touches the n = 1 − bend and abifurcation of higher codimension forms at the point of contact, see Fig. 13 (b). It is noteworthythat the second primary bifurcation occurs at exactly the same value of a as the high-codimensionpoint. Caused by the complete decoupling at ∆ = 0 , φ is exactly zero on the complete n =1 + branch. Furthermore the eigenvalue λ − = λ does not depend on the φ -component of thecorresponding steady state. Therefore, the second primary bifurcation and the first secondarybifurcation occur at identical a . This implies that at smaller a there exist many further pairs andeven groups of simultaneous bifurcations. However, the inset of Fig. 13 (b) shows that the dottedline connecting the two bifurcations is not exact vertical. Instead it bifurcates supercritically (i.e.,to the left), folds to the right in a saddle-node bifurcation before becoming vertical again at the39 Preprint– contact: t [email protected] – October 5, 2020 rossing point that coincides with its second saddle-node bifurcation.Slightly increasing α further, we obtain the diagram in Fig. 13 (c) that shows very rich behaviorin the region of the crossing point in Fig. 13 (b). Now, the two primary bifurcations are directlylinked by an n = 1 branch of steady states. The n = 1 + part emerges supercritically and becomesunstable via a secondary drift pitchfork bifurcation exactly at the apex of the branch. FollowingRef. [59] one can derive a condition, (cid:82) φ + ρ + αρ − α φ d x , for drift pitchfork bifurcations tooccur. The n = 1 − part emerges subcritically since the chosen parameters correspond to a locusinside the orange shaded region of Fig. 11. In addition, it features a secondary Hopf bifurcation.The branch of traveling n = 1 states that emerges at the drift pitchfork bifurcation is first linearlystable [cf. Fig. 14 (a)], then destabilizes in a Hopf bifurcation before finally ending in anotherdrift pitchfork bifurcation on the unstable part of the “upper left part” of the steady n = 1 branch(green dotted line). The latter one then stabilizes in a saddle-node bifurcation at a ≈ − . (greensolid line). Tracking the branches of time-periodic states emerging at the Hopf bifurcations untiltheir termination is numerically rather challenging. Therefore we accompany the continuationresults with results of direct time simulations [marked by bold “+”-symbols in inset of panel (c)].Fig. 14 shows a selection of space-time plots which illustrate the various qualitatively differentbehaviors at different values of a . The time evolutions are initialized with a noisy homogeneousstate. Drawing on both sets of results proposes the following bifurcation behavior: At the Hopfbifurcation on the stationary n = 1 branch (blue line) a branch of standing waves (red dotted line)emerges supercritically, i.e., towards smaller a , and carries one unstable eigenvalue. A branch ofmodulated waves (light blue line) emerges supercritically at the Hopf bifurcation of the travelingwave state (gray line) and is at first stable. An example of such a state is given in Fig. 14 (b).The magnification in Fig. 13 (c) focuses on the region where both branches of time-periodic statesapproach each other. Taking results from continuation and time simulations into account onecan discern that the branch of modulated waves terminate on the branch of standing waves at a ≈ − . . At the corresponding drift bifurcation, the standing waves gain stability [transitionfrom dotted to solid line, cf. Fig. 14 (c)]. The corresponding branch continues toward a globalhomoclinic bifurcation on the unstable part of the n = 1 branch of steady states (green dottedline). In particular, we find a narrow window of multistability of standing waves and steady states.A further increase of α , gives Fig. 13 (d), where the half-loop of n = 1 states connected to theprimary bifurcations has shrunk. Note that we do not show the time-periodic states. With furtherincreasing α the two primary pitchfork bifurcations move closer together and eventually fuse into40 Preprint– contact: t [email protected] – October 5, 2020 primary Hopf bifurcation when the two eigenvalues form a complex conjugate pair. The resultis a bifurcation diagram as in Fig. 13 (e), where the branch of traveling states directly emerges ina primary Hopf bifurcation. Note that the transition between the structure of primary bifurcationsin Figs. 13 (d) and (e) is also of higher codimension, as more than two bifurcations fuse to becomethe Hopf bifurcation.Finally, we briefly discuss the high codimension point in Fig. 13 (b): If all the structure de-scribed for Fig. 13 (c) emerges at the high-codimension point of Fig. 13 (b), only consideringsecondary bifurcations this point “contains” one Bogdanov-Takens bifurcations, a double driftpitchfork bifurcation and an inverse necking bifurcation, i.e., three standard codimension-2 bifur-cations. To test this, we present in Fig. 13 (f) the loci of all local secondary bifurcations visible inFig. 13 (c) in the ( a, α )-plane. They are obtained by two-parameter continuations. Indeed, it indi-cates that all five tracked bifurcations emerge from the single point of high codimension markedby the square symbol in Fig. 13 (b). The reason lies again in the symmetries of the consideredspecial case with ¯ φ = ¯ φ = 0 .To next consider scenario B , in agreement with condition (27), we need to change the sign of M or of α : In Fig. 15 we use M < and decrease the nonvariational coupling in two steps from α > − ρ to α < − ρ while keeping the remaining parameters as in Fig. 13. We find, that in contrastto the rich transition behavior in scenario A , scenario B is rather dull.In Fig. 15 (a) for α = − ρ = − . the systems shows a stationary large-scale instability andboth, n = 1 + and n = 1 − , branches emerge supercritically. As ∆ = 0 , again one field isdecoupled, here it is φ (due to the switched sign of α ). In contrast to scenario A , where the n + branch is characterized by φ = 0 , here the n − branch features a zero φ -field. Thus, the argumentfor the simultaneous occurrence of a pair of bifurcations on the trivial branch and the n = 1 + branch does not apply. Instead the n = 1 + branch stays stable and no point of higher codimensionappears. Decreasing α , the primary bifurcations approach each other, see Fig. 15 (b). Furthermorethe system reaches the blue shaded region in Fig. 11 resulting in subcritical behavior of the n = 1 + branch.Finally, the primary bifurcations collide at the Hopf threshold [see Eq. (B2)] and with furtherdecreasing α a branch of time-dependent states emerges not unlike a zipper. It connects the pri-mary Hopf bifurcation via a branch of stationary traveling states with the steady n = 1 branchwhere it ends in a drift pitchfork bifurcations. There is a further Hopf bifurcation where a branchof modulated traveling states emerges (not shown). In summary, in scenario B stable time-periodic41 Preprint– contact: t [email protected] – October 5, 2020 . − . − . − . − . − . − . a . . . . . . || δ φ || (a) n =1 + n =1 − − . − . − . − . − . − . − . a . . . . . . || δ φ || (b) n =1 + n =1 − − . − . − . − . − . − . − . a . . . . . . || δ φ || (c) FIG. 15:
Panels (a)-(c) show a sequence of bifurcation diagrams that illustrates scenario B of the emer-gence of subcriticality and subsequent time-periodic behavior in the fully nonlinear regime for decreasingnonvariational coupling α = − . , − . and − . at ρ = 1 . and M < . Linestyles, symbols andremaining parameters are as in Fig. 13. behavior only arises when the primary pitchfork bifurcations collide at the onset of a large-scaleoscillatory instability and the emergence of the related Hopf bifurcation.Note that in the case of purely nonvariational coupling time-periodic behavior may occur atarbitrarily small nonvariational coupling. This remarkable case is analyzed in appendix D.42 Preprint– contact: t [email protected] – October 5, 2020 . CONCLUSION
We have systematically analyzed the influence of nonvariational (or active, or non-reciprocal)coupling in a two-field model for phase separation in a binary/ternary mixture. Our study has onthe one hand shed light on the transition from large-scale demixing (mediated by coarsening) to theformation of patterns. On the other hand it has highlighted the transition from stationary states totime-periodic behavior that occurs when increasing activity. In particular, we have studied a CH-type mass-conserving dynamics of two linearly coupled concentration fields. We emphasize thatthe coupling terms do not affect the conservation properties, i.e. both fields stay conserved. Thisis in contrast to the coupling used in Ref. [41]. The kept conservation property for both speciesmakes the model well suited to describe the dynamics of different chemical or biological entitiesthat show non-reciprocal interactions but do not transform into each other or otherwise changetheir number on the considered time scales. This includes catalytic species whose interaction ismediated via other species not explicitly described be the model, or bacterial populations with apredator-prey type attraction-repulsion pattern. For a microscopic model of a similar chemicalsystem see Ref. [60].The employed linear coupling between the two species corresponds to cross-diffusion terms andis composed of a symmetric (variational) and an asymmetric (nonvariational) term. The asymmet-ric coupling represents the only active element of the model. It is often called “nonreciprocal” asit breaks the third law of Newtonian mechanics [61].The linear stability analysis in Sec. II provides a surprising rich linear behavior [cf. Fig. 1] in-cluding particularly Turing and large-scale Hopf-instabilities if the nonvariational coupling dom-inates the variational one ( | α | > | ρ | ). It is notable that a relatively simple mapping to the linearstability analysis of the classical Turing system of coupled reaction-diffusion (RD) equations [53]can be given and allows to draw parallels between the respective parameters. Most importantly,the ratio of rigidities κ takes the role of the ratio of diffusion constants in the RD system implyingthat much of the more intricate behavior can only be found if κ (cid:54) = 1 .In particular, κ (cid:54) = 1 is a necessary condition for a Turing instability to be found. Here, we havefocused on the related transition from large- to small-scale instability and the resulting changes incoarsening behavior. We have provided an analysis based on the bifurcation behavior for smallsystems. For larger systems such an instability implies that features have to be expected likethe emergence of localized states in a slanted snakes-and-ladders structure [62, 63]. This will be43 Preprint– contact: t [email protected] – October 5, 2020 ursued elsewhere [64].We have then briefly treated the passive limit of the model where well-studied phase separationand coarsening dynamics characterizes the system that ultimately approaches a equilibrium state.We have related common phase diagrams to corresponding bifurcation diagrams which have thenserved as the main tool to discuss the behavior in the nonvariational case, too.The main results of our work has been presented in Sec. IV where the effects of nonvaria-tional coupling have been studied. We have described three different mechanisms of suppressionof coarsening, which we term (i) linear suppression , (ii) nonlinear suppression and (iii) nonlineararrest of coarsening and have shown how the linear suppression relates to the Turing instabil-ity and the corresponding primary bifurcations and how nonlinear suppression and arrest relateto secondary bifurcations where particular patterns stabilize. Interestingly, neither a Turing in-stability nor a suppression of coarsening is reported in recent studies of the seemingly identicalmodel [39, 40]. Instead, suppression of coarsening is found in two-field CH-type models wherethe coupling breaks the conservation property [41, 47]. This could result in the hypothesis that thesuppression of coarsening for CH-type models necessarily needs a non-conserved dynamics. Notethat a similar transition is described by a thin-film model of a Rayleigh-Taylor unstable heatedevaporating liquid film [65]. The term that drives the transition in this CH-type equation one maycall “nonvariational evaporation”.Our findings, however, clearly indicates that such a hypothesis would not be correct: coarseningcan also be stopped by nonvariational coupling within a fully conserved dynamics. Our analysishas shown that the subtle differences between our model and the literature models have dramaticconsequences. Namely, in our notation Ref. [39] considers the non-generic case of equal rigidities( κ = 1 ). In this special case Turing instability and all secondary bifurcations are absent. In con-trast, Ref. [40] mostly investigates a limiting case where one rigidity is identical zero ( κ = 0 ). Ina small part they too use the non-generic case of equal rigidities. In consequence, suppression ofcoarsening can not occur. One can conclude, that suppression of coarsening can be promoted by(i) breaking the conservation property of the dynamics while preserving the variational character[47] or (ii) by nonvariational additions, e.g., the presently studied linear non-reciprocal interac-tions, while keeping a conserved dynamics for all fields. Future work should investigate if thereare qualitative differences between these two cases, e.g., regarding the prevalence of linear vs.nonlinear mechanisms of coarsening or systematic changes to underlying scaling laws [42].An analysis of the case of subcritical primary bifurcations has revealed another intriguing fea-44 Preprint– contact: t [email protected] – October 5, 2020 ure of the model. Based on a weakly nonlinear analysis (see Appendix C) we have shown that,surprisingly, nonvariational coupling can cause subcriticality even at zero mean concentrationswhere symmetry arguments indicate its nonexistence. Normally, one expects a subcritical charac-ter of phase separation only beyond a nonzero critical mean concentration. However, caused by thenon-reciprocal character of the contribution of activity the common understanding of subcriticalitydoes not apply.Finally, we have focused on the emergence of time-dependent states, for simplicity focusingon the case of zero mean concentrations. We have described two distinguished scenarios for theemergence that appear to be rather generic for nonvariational models. On the one hand, travelingor standing waves can directly emerge from the trivial homogeneous state when the latter becomesunstable with respect to a Hopf bifurcation. On the other hand, when the trivial state is unstablewith respect to a stationary mode, first a inhomogeneous steady state emerges that starts to travelafter a secondary drift-pitchfork bifurcation occurs. Similar behavior is also found for other activemodels, e.g., an active phase-field-crystal model [59]. At the drift-pitchfork bifurcation of theparity-symmetric state, the symmetry is broken and the newly emerging state drifts with a velocitythat shows square-root behavior as determined in a one-mode approximation, e.g., in Ref. [40] forthe closely related model studied there. Further secondary Hopf and drift-pitchfork bifurcationscreate a rich variety of time-periodic states and multistable regions exist where steady and time-dependent states are both linearly stable.A limitation of our study is the simple linear coupling of the two fields. It is reasonable forweakly coupled fields, but needs to be amended for strong coupling. In the future, it might then beinteresting to study how the interplay of nonlinear variational and nonvariational couplings alterthe observed behavior. 45
Preprint– contact: t [email protected] – October 5, 2020 ppendix A: Nondimensionalisation
This appendix discusses our nondimensionalization of the coupled CH model and thereby elu-cidates the physical meaning of the various nondimensional parameters of the model (4). We startwith the dimensional coupled system ∂φ ∂t = Q ∂ ∂x (cid:18) − κ ∂ φ ∂x + ζ f (cid:48) ( φ ) − γ φ (cid:19) ∂φ ∂t = Q ∂ ∂x (cid:18) − κ ∂ φ ∂x + ζ f (cid:48) ( φ ) − γ φ (cid:19) , (A1)and express the dimensional fields as φ = ˆ φ ˜ φ and φ = ˆ φ ˜ φ , where a hat indicates a fixed scale(to be determined) and a tilde the nondimensional quantity. Furthermore we introduce general en-ergy scales via f i = ˆ f i ˜ f i , a characteristic time [length] scale τ [ L ] via t = τ ˜ t [ x = ˆ L ˜ x = L(cid:96) ˜ x ].Here ˆ L is the physical domain size. For practical reasons we introduce the (nondimensional) ratio L = ˆ L(cid:96) with (cid:96) being the computational domain size. In this way we can fix the computationaldomain size (cid:96) = 1 and use the dimensionless ratio L to control the physical domain size. After in-troducing the nondimensional quantities and re-grouping parameters, the nondimensional systemof equations is ∂∂ ˜ t ˜ φ = Q τ κ L ∂ ∂ ˜ x (cid:32) − L ∂ ˜ φ ∂ ˜ x + ζ ˆ φ κ ˆ f ˜ f (cid:48) ( ˆ φ ˜ φ ) − ˆ φ γ ˆ φ κ ˜ φ (cid:33) ∂∂ ˜ t ˜ φ = Q Q Q τ κ L ∂ ∂ ˜ x (cid:32) − κ κ L ∂ ˜ φ ∂ ˜ x + ζ ˆ φ κ ˆ f ˜ f (cid:48) ( ˆ φ ˜ φ ) − ˆ φ γ ˆ φ κ ˜ φ (cid:33) . (A2)It contains nondimensional combinations of physical parameters and of the scales τ, ˆ φ , and ˆ φ that still need to be chosen.We define Q ≡ Q Q , ˜ γ ≡ ˆ φ γ ˆ φ κ , ˜ γ ≡ ˆ φ γ ˆ φ κ , κ ≡ κ κ , ˜ a ≡ ζ ˆ f κ , ˜ a ≡ ζ ˆ f κ . (A3)set τ = 1 /Q κ , and obtain ∂∂ ˜ t ˜ φ = 1 L ∂ ∂ ˜ x (cid:32) − L ∂ ˜ φ ∂ ˜ x + ˜ a ˆ φ f (cid:48) ( ˆ φ ˜ φ ) − ˜ γ ˜ φ (cid:33) ∂∂ ˜ t ˜ φ = QL ∂ ∂ ˜ x (cid:32) − κL ∂ ˜ φ ∂ ˜ x + ˜ a ˆ φ g (cid:48) ( ˆ φ ˜ φ ) − ˜ γ ˜ φ (cid:33) . (A4)We assume the bulk energies to be double-well potentials, i.e. the derivatives are f (cid:48) ( ˆ φ ˜ φ ) = (cid:16) b ˆ φ ˜ φ (cid:17) ˆ φ ˜ φ (A5) g (cid:48) ( ˆ φ ˜ φ ) = (cid:16) b ˆ φ ˜ φ (cid:17) ˆ φ ˜ φ . (A6)46 Preprint– contact: t [email protected] – October 5, 2020 e set ˆ φ = (cid:114) b ˜ a , ˆ φ = (cid:114) b ˜ a (A7)and obtain ∂∂ ˜ t ˜ φ = 1 L ∂ ∂ ˜ x (cid:32) − L ∂ ˜ φ ∂ ˜ x + ˜ a ˜ φ + ˜ φ − ˜ γ ˜ φ (cid:33) ∂∂ ˜ t ˜ φ = QL ∂ ∂ ˜ x (cid:32) − κL ∂ ˜ φ ∂ ˜ x + ˜ a ˜ φ + ˜ φ − ˜ γ ˜ φ (cid:33) (A8)The parameter a of the linear term in the single CH equation ( ∼ ∂ xx aφ ) is often referred to astemperature. Therefore we set ˜ a = ˜ a + a ∆ ≡ a + a ∆ and use a as an effective temperature.Then a ∆ represents the shift in critical temperature between the two uncoupled CH phase dia-grams. Furthermore we split the two coupling parameters into the symmetric and antisymmetriccontributions, that physically represent variational (reciprocal) and nonvariational (nonreciprocal)coupling, respectively. Namely, ρ = ˜ γ + ˜ γ α = ˜ γ − ˜ γ Dropping the tildes we obtain the nondimensionalised system ∂∂t φ = 1 L ∂ ∂x (cid:18) − L ∂ φ ∂x + f (cid:48) ( φ ) − ( ρ + α ) φ (cid:19) ∂∂t φ = QL ∂ ∂x (cid:18) − κL ∂ φ ∂x + f (cid:48) ( φ ) − ( ρ − α ) φ (cid:19) (A9)with f (cid:48) ( φ ) = aφ + φ and f (cid:48) ( φ ) = ( a + a ∆ ) φ + φ . Appendix B: Linear stability results for specific f i In Section II we have analyzed the linear stability of homogeneous states for the model pre-sented in Eqs. (4). The stability diagrams in Fig. 1 summarize the general linear results. Here wespecify them for our case where f (cid:48)(cid:48) = a + 3 ¯ φ , f (cid:48)(cid:48) = a + a ∆ + 3 ¯ φ , Q = 1 and use a as maincontrol parameter. Note that we use the abbreviation q = k/L throughout the appendix. First, the47 Preprint– contact: t [email protected] – October 5, 2020 ispersion relations (12) become λ ± ( q ) = 12 q (cid:40) − (cid:104) q (1 + κ ) + 2 a + a ∆ + 3 (cid:16) φ + φ (cid:17)(cid:105) ± (cid:114)(cid:104) q (1 − κ ) + 3 (cid:16) φ − φ (cid:17) − a ∆ (cid:105) − (cid:41) . (B1)Since the discriminant is independent of a , it does not influence the occurrence of complex eigen-values. In contrast, the coupling strengths ρ and α only appear in the combination ∆ = α − ρ and only enter the discriminant. Complex eigenvalues occur if ∆ > (cid:104) q (1 − κ ) + 3 (cid:16) φ − φ (cid:17) − a ∆ (cid:105) (B2)Then, Hopf bifurcations of modes with wavenumber q occur at [cf. (13)] a o ( q ) = − (cid:2) q (1 + κ ) + a ∆ + 3 (cid:0) ¯ φ + ¯ φ (cid:1)(cid:3) (B3)independently of both coupling strengths. In consequence, the onset of the large-scale oscillatoryinstability occurs at a H = a o (0) = − (cid:2) a ∆ + 3 (cid:0) ¯ φ + ¯ φ (cid:1)(cid:3) . (B4)For stationary instabilities we use Eq. (16) to obtain the critical values a ± ( q ) = 12 (cid:40) − (cid:2) q (1 + κ ) + a ∆ + 3 (cid:0) ¯ φ + ¯ φ (cid:1)(cid:3) ± (cid:113)(cid:2) q (1 − κ ) + 3 (cid:0) ¯ φ − ¯ φ (cid:1) − a ∆ (cid:3) − (cid:27) . (B5)Since the dispersion relation has two branches, λ ± ( q ) , we obtain two critical a . The stabilityborder in the ( q, a ) -plane is then represented by the a + ( q ) curve where λ + ( q ) changes sign. Inparticular, the onset of a large-scale stationary instability is at a CH = a + ( q c = 0) = 12 (cid:26) − (cid:2) a ∆ + 3 (cid:0) ¯ φ + ¯ φ (cid:1)(cid:3) + (cid:113)(cid:2) (cid:0) ¯ φ − ¯ φ (cid:1) − a ∆ (cid:3) − (cid:27) . (B6)For a small-scale stationary instability the onset is at nonzero q c [cf. Eq. (17)] at a T ≡ a + ( q c ) = 1 κ − (cid:104) a ∆ ∓ √ κ ∆ (cid:105) (B7)[cf. Eq. (19)]. Stability borders a o ( q ) [Eq. (B3)] for oscillatory and a + ( q ) [Eq. (B5)] for stationaryinstabilities are plotted for different cases in Figs.2, 3 and 4.48 Preprint– contact: t [email protected] – October 5, 2020 n alternative approach is to consider the shape of the dispersion relations [Eq. (B1)] at fixedparameters. For q = 0 both eigenvalues λ ± are always zero as expected for two conservation laws.Setting λ ± = 0 , we can determine other real roots. Due to left-right symmetry, there are 0, 1 or2 nonzero and positive wavenumber values q ± = (cid:115) − f (cid:48)(cid:48) − f (cid:48)(cid:48) κ ± (cid:112) ( f (cid:48)(cid:48) − κf (cid:48)(cid:48) ) − κ ∆2 κ , (B8)with f (cid:48)(cid:48) and f (cid:48)(cid:48) as given above, where the dispersion relation crosses zero. If q + becomes realit corresponds to a nontrivial root of λ + (note that λ + > λ − ). If both, q + and q − are real, twonontrivial roots exist. Again q + corresponds to a root of λ + . The second root q − can correspondto a root of λ − or to a second root of λ + . The latter case corresponds to the appearance of asmall-scale instability. An oscillatory instability has its threshold when the real part of λ ± ( q ) iszero. This gives 0 or 1 nonzero and positive wavenumber values q o = (cid:115) − a + a ∆ + 3 (cid:0) ¯ φ + ¯ φ (cid:1) κ , (B9)referring that only small-scale oscillatory instabilities appear.Independently of the onset of instabilities we can determine the band of wavenumbers (cid:2) q o − , q o + (cid:3) where complex eigenvalues occur by setting the discriminant in Eq. (B1) to zero. This yields q o ± = (cid:115) ± √ ∆ − (cid:0) φ − φ − a ∆ (cid:1) − κ . (B10)Requesting that the limiting values q o ± have to be real implies that for φ − φ − a ∆ > √ ∆ − √ ∆ < φ − φ − a ∆ < √ ∆3 ¯ φ − φ − a ∆ < − √ ∆ complex eigenvalues occur in the bands (cid:2) q o − , q o + (cid:3) if κ > (cid:2) , q o ± (cid:3) if κ ≷ (cid:2) q o − , q o + (cid:3) if κ < . (B11)In the special case of equal interface rigidity ( κ = 1 ), the occurrence of complex eigenvalues isindependent of the wavenumber as q o ± → ∞ for κ → . Then eigenvalues are complex at any q for (cid:2)
3( ¯ φ − ¯ φ ) − a ∆ (cid:3) < . (B12)49 Preprint– contact: t [email protected] – October 5, 2020 ppendix C: Weakly nonlinear analysis
To identify parameter values where the primary pitchfork bifurcations change character fromsupercitical to subcritical we apply weakly nonlinear analysis. Although we are mainly interestedin the onset of subcritical behavior for zero mean concentrations we develop the theory for thegeneral case by employing the following ansatz φ = ¯ φ + (cid:112) | µ | A e iq n Lx + | µ | C e iq n Lx + c.c. (C1) with A = v A + | µ | A (C2)with µ is the smallness parameter that defines the distance to the bifurcation at µ = µ c = 0 withrescaled, discretized wavenumber q = q n = nπL . We should note that we do not consider spaceand time dependent amplitudes, i.e. we do not derive any partial differential amplitude equation.Thus, we do not need any Fourier mode ∼ e since the mean concentrations are fixed by ¯ φ and cannot be altered through the dynamics due to the mass conserving property of the system. Besides,the bifurcation points are simple codimension-1-points, i.e. they are represented by lines in the ( f (cid:48)(cid:48) , f (cid:48)(cid:48) ) -plane [see e.g. green and orange lines in Fig. 1] where f (cid:48)(cid:48) and f (cid:48)(cid:48) are taken at ¯ φ and ¯ φ , respectively. Nevertheless, we use both f (cid:48)(cid:48) and f (cid:48)(cid:48) as control parameters in order to keep ourresult more general. Hence, one of the two parameters can be chosen arbitrarily in some range,then the other one is adjusted to the corresponding line of the bifurcation at ( f (cid:48)(cid:48) , f (cid:48)(cid:48) ) = ( f (cid:48)(cid:48) , c , f (cid:48)(cid:48) , c ) .In contrast to Fig. 1 where the colored lines represent the onset of linear instability, here weconsider an arbitrary primary bifurcation with fixed wavenumber q n . Hence, the critical parametersdepend on the given wavenumber, i.e. f (cid:48)(cid:48) i, c = f (cid:48)(cid:48) i, c ( q n ) . In the following we do not indicate thisexplicit dependency. Starting from the bifurcation we can then choose any arbitrary direction inthe ( f (cid:48)(cid:48) , f (cid:48)(cid:48) ) -plane by introducing an angle ϑ ∈ [0 , π [ (and µ > or µ < ) and f (cid:48)(cid:48) = f (cid:48)(cid:48) , c + µ sin ϑ , f (cid:48)(cid:48) = f (cid:48)(cid:48) , c + µ cos ϑ . (C3)The eigenvector v solves the linear equation L (cid:12)(cid:12)(cid:12) q n ,µ c v = q n + f (cid:48)(cid:48) , c − ( ρ + α ) − ( ρ − α ) κq n + f (cid:48)(cid:48) , c v = 0 . (C4)If one considers the onset of linear instability in a Turing bifurcation for arbitrary f (cid:48)(cid:48) , then f (cid:48)(cid:48) , c = f (cid:48)(cid:48) T [Eq. (19)] and q n = k c /L [Eq. (17)]. Our calculations, however, hold for any stationary50 Preprint– contact: t [email protected] – October 5, 2020 rimary bifurcation even far away from the onset of linear instability. The eigenvector v and theadjoint eigenvector v † are given by v = ρ + αq n + f (cid:48)(cid:48) , c , v † = ρ − αq n + f (cid:48)(cid:48) , c . (C5)The latter one forms the kernel of the adjoint linear operator L † (cid:12)(cid:12)(cid:12) q n ,µ c and will be used to applythe Fredholm alternative. We insert ansatz (C1) into the model Eqs. (4) (for Q = 1 ), compare theFourier coefficients and sort different orders of | µ | , then we obtain at O ( | µ | ) the coefficients of e iq n Lx : | µ | L (cid:12)(cid:12)(cid:12) q n ,µ c C = 12! f (cid:48)(cid:48)(cid:48) v f (cid:48)(cid:48)(cid:48) v | µ | A ≡ a | µ | A (C6) ⇒ C = L a − L a L L − L L (cid:12)(cid:12)(cid:12) q n ,µ c (cid:124) (cid:123)(cid:122) (cid:125) ≡ γ A , C = L a − L a L L − L L (cid:12)(cid:12)(cid:12) q n ,µ c (cid:124) (cid:123)(cid:122) (cid:125) ≡ γ A (C7)with v [ v ] being the first [second] component of the eigenvector v . At order O ( (cid:112) | µ | ) wecompare the coefficients of the mode e iq n Lx : | µ | / L (cid:12)(cid:12)(cid:12) q n ,µ c A + ∂∂µ L (cid:12)(cid:12)(cid:12) q n ,µ c v µ | µ | / A + 22! f (cid:48)(cid:48)(cid:48) v C f (cid:48)(cid:48)(cid:48) v C | µ | / A ∗ + 13! f (cid:48)(cid:48)(cid:48)(cid:48) v f (cid:48)(cid:48)(cid:48)(cid:48) v | µ | / | A | A = 0 (C8)We apply the Fredholm alternative, i.e. we multiply Eq. (C8) with v † from the left. Then the firstterm vanishes. We insert the expressions for amplitudes C and C [Eq. (C7)], we reincorporatethe smallness parameter into the amplitude A and finally with ∂∂µ L (cid:12)(cid:12)(cid:12) q n ,µ c v = v sin ϑv cos ϑ weobtain the stationary amplitude equation µ (cid:16) v v † sin ϑ + v v † cos ϑ (cid:17) A + 22! (cid:16) f (cid:48)(cid:48)(cid:48) v v † γ + f (cid:48)(cid:48)(cid:48) v v † γ (cid:17) | A | A + 33! (cid:16) f (cid:48)(cid:48)(cid:48)(cid:48) v v † + f (cid:48)(cid:48)(cid:48)(cid:48) v v † (cid:17) | A | A = 0 (C9)First we reproduce the result for the standard CH equation as the decoupled limit, i.e. for ρ = α = 0 . Then the expressions turn out to be rather simple with v = v † = γ = 0 , v = v † = 1 and γ = a L = f (cid:48)(cid:48)(cid:48) ( q n + f (cid:48)(cid:48) , c ) . In the decoupled case one control parameter does not influence the51 Preprint– contact: t [email protected] – October 5, 2020 tability, here f (cid:48)(cid:48) , i.e. the only possible direction is ϑ = 0 . The stationary amplitude equation[Eq. (C9)] reduces to µA + (cid:34) f (cid:48)(cid:48)(cid:48) (cid:0) q n + f (cid:48)(cid:48) , c (cid:1) + f (cid:48)(cid:48)(cid:48)(cid:48) (cid:35) | A | A = 0 (C10)Next we discuss the case of coupled fields with zero mean concentrations, i.e. ¯ φ = (0 , . Withthe standard double-well potentials f i ∼ φ i + φ i the third derivatives taken at ¯ φ vanish since f (cid:48)(cid:48)(cid:48) i ∼ ¯ φ i = 0 . In this case no Fourier mode ∼ e iq n Lx is excited, i.e. γ , = 0 . The generalstationary amplitude equation [Eq. (C9)] reduces to µ (cid:16) v v † sin ϑ + v v † cos ϑ (cid:17) A + 33! (cid:16) f (cid:48)(cid:48)(cid:48)(cid:48) v v † + f (cid:48)(cid:48)(cid:48)(cid:48) v v † (cid:17) | A | A = 0 (C11)Inserting the components of the (adjoint) eigenvectors we obtain µ (cid:16)(cid:0) ρ − α (cid:1) sin ϑ + (cid:0) q n + f (cid:48)(cid:48) , c (cid:1) cos ϑ (cid:17) A + 33! (cid:16) f (cid:48)(cid:48)(cid:48)(cid:48) (cid:0) ρ − α (cid:1) ( ρ + α ) + f (cid:48)(cid:48)(cid:48)(cid:48) (cid:0) q n + f (cid:48)(cid:48) , c (cid:1) (cid:17) | A | A = 0 (C12)We can choose the amplitude A to be real and positive (since we use periodic boundary condi-tions) and solve the amplitude equation which yields A = 0 A = (cid:115) − µ ( ρ + α ) ( ρ − α ) sin ϑ + ξ cos ϑf (cid:48)(cid:48)(cid:48)(cid:48) ( ρ − α ) + f (cid:48)(cid:48)(cid:48)(cid:48) ξ ( ρ + α ) (C13)where we introduce the abbreviation ξ = (cid:0) q n + f (cid:48)(cid:48) , c (cid:1) . (C14)Throughout this work the bifurcation diagrams are calculated for diagonal cuts through the stabil-ity plane, i.e. f (cid:48)(cid:48) = a f (cid:48)(cid:48) = a + a ∆ f (cid:48)(cid:48)(cid:48)(cid:48) = f (cid:48)(cid:48)(cid:48)(cid:48) = 6 ϑ = π/ (C15) ⇒ A = (cid:115) − ˜ µ ( ρ − α ) + ξ ( ρ − α ) + ξ ( ρ + α ) with ˜ µ = µ √ ρ + α ) . (C16)Then the critical parameter at given wavenumber q n is given by f (cid:48)(cid:48) , c = a ± ( q n ) [see Eq. (B5)] where + [ − ] refers to the upper [lower] eigenvalue λ + [ λ − ]. Furthermore we obtain ξ ± = (cid:0) q n + a ± ( q n ) (cid:1) = (cid:32) − M ± (cid:114) M ρ − α ) (cid:33) with M = q n (1 − κ ) − a ∆ . (C17)52 Preprint– contact: t [email protected] – October 5, 2020 he trivial state ( A = 0 ) looses stability for decreasing a , i.e. for ˜ µ < . Then the correspondingbifurcation is subcritical if ( ρ − α ) + ξ ( ρ − α ) + ξ ( ρ + α ) < (C18) ⇒ − max (cid:18) ξ , ξ ( ρ + α ) (cid:19) < ρ − α < − min (cid:18) ξ , ξ ( ρ + α ) (cid:19) (C19)This leads to the necessary condition | α | > | ρ | , i.e. the nonvariational coupling needs to bestronger than the variational one in order to induce subcritical behavior with zero concentrations.Due to ξ ( M ) = ξ − ( − M ) branches of same periodicity n related to λ + and λ − , respectively,exchange subcritical behavior if M switches sign. For M = 0 and ρ = 0 no subcritical behavioroccurs. Especially for κ = 1 parameter ξ = − a ∆ ± (cid:113) a + ( ρ − α ) is independent of thewavenumber, then there exists one subcriticality condition for all stationary primary bifurcation.The criterion [Eq. (C19)] is illustrated in Fig. 11. Further discussion, especially its impact on theonset of time-periodic behavior is discussed in Secs. IV B and IV C. Appendix D: Time-periodic behavior arbitrarily close to equilibrium
In Sec. IV C we have analyzed two generic scenarios of the emergence of time-periodic behav-ior. Here, we briefly investigate the particular case of purely nonvariational coupling, i.e., ρ = 0 , α (cid:54) = 0 . This is a very special situation as the necessary condition for time-periodic behavior | α | > | ρ | is fulfilled at arbitrarily small α . Therefore, one might expect to find oscillatory behaviorarbitrarily close to the classical gradient dynamics case that describes systems evolving towardsthermodynamic equilibrium. If the uncoupled subsystems are equal, i.e., if f (cid:48)(cid:48) = f (cid:48)(cid:48) and κ = 1 ,all primary pitchfork bifurcations turn into Hopf bifurcations for any nonzero α . This, however, isa highly non-generic case due to the specific condition of equal subsystems. Furthermore, time-periodic states only exist with small amplitude (proof not shown).Significantly more relevant is the case of unequal subsystems shown in Fig. 16 for a smallnonvariational coupling α = 0 . . The bifurcation diagram in panel (a) shows the uniform stateand the stable parts of three different phase-separated states. Their unstable parts are omitted sincethe branches could not be distinguished by eye. The uniform state (black line) looses stability in alarge-scale instability where the stationary n = 1 branch (green line) emerges supercritically, andhence, stable. Panel (d) illustrates the emerging state and shows that the corresponding fields are53 Preprint– contact: t [email protected] – October 5, 2020 n-phase, i.e., the nonvariational coupling acts attracting near the onset of linear instability. Far inthe nonlinear regime, however, the system favors anti-phase fields so that for a (cid:46) − . the stateshown in panel (b) is stable (green line). The stable, stationary one-periodic states with in-phaseand anti-phase fields, respectively, are connected via a branch of stable, drifting states (gray line)which originates from drift pitchfork bifurcations (triangles). The example profile of one of thesedrifting states in panel (c) indicates the phase shift from in-phase to anti-phase along the branch.For any phase unequal and π the states drift with constant velocity which is indicated by anarrow in panel (c). − . − . − . a . . . . . . . . || δ φ || (a) − . . . − . . .
25 (d) φ φ − . . . −
101 (c) −→ v − . . . −
101 (b)
FIG. 16:
Emergence of drifting states close to equilibrium for α = 0 . . Panel (a) shows a large scalestationary instability from the uniform state (black line). The loci of the selected space-time plots in panels(b)-(d) are marked by bold “+”- symbols in (a). Stable stationary n = 1 branches occur for in-phase fields[panel (d), blue line in panel (a)] and anti-phase fields [panel (b), green line in (a)]. A branch of stabledrifting states [panel (c), gray line in (a)] connects the stationary ones. The remaining parameters are a ∆ = − . , κ = 2 . , ¯ φ = 0 . , ¯ φ = 0 . , L = 4 π and ρ = 0 . In addition to the unique property of time-periodic behavior arbitrarily close to equilibrium, the54
Preprint– contact: t [email protected] – October 5, 2020 urely nonvariational coupling also represents a special case regarding model classification: For ρ = 0 , we can write Eq. (4) in gradient dynamics form ∂ t φ i = ∂ x (cid:32) Q i L ∂ x δ (cid:101) F δφ i (cid:33) i = 1 , (D1)with (cid:101) F = (cid:90) (cid:20) − L |∇ φ | − f ( φ ) + κ L |∇ φ | + f ( φ ) + αφ φ (cid:21) d x (D2)and Q = − , Q = Q . (D3)In this formulation the energy has destabilizing and stabilizing gradient-square terms. However,the active character is then encoded in the negative mobility constant Q .55 Preprint– contact: t [email protected] – October 5, 2020
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