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coarsening has been a object of intensive study over the last decades not only because of its experimental relevance but also because it constitutes what is maybe the simplest case of out - of - equilibrium dynamics . for a scalar order parameter , one generally distinguishes the non - conserved case , e.g. the ising model with glauber dynamics ( model a according to the widely - used classification of @xcite ) , from the conserved case , e.g. the ising model with kawasaki dynamics ( model b ) . it is now well - established that in these cases phase ordering is characterized by a single lengthscale @xmath3 growing algebraically with time ( @xmath4 ) with the growth exponent @xmath5 taking two different values , @xmath1 for the non - conserved / model  a case , and @xmath0 for conserved order parameter systems like model  b @xcite . here we are interested in the more complicated and therefore less well understood case of a non - conserved order parameter coupled to a conserved concentration ( so - called model  c ) . examples of this situation can be found in various physical systems , e.g. , intermetallic alloys ( see @xcite and references therein ) , adsorbed layers on solid substrates @xcite and supercooled liquids @xcite . in these systems , the decomposition process ( described by the conserved concentration @xmath6 ) and the ordering process ( described by the non - conserved order parameter @xmath7 ) are coupled . let us consider an @xmath8 alloy on a square lattice in order to illustrate this . a state in which all @xmath9 atoms are surrounded by @xmath10 atoms is energetically favorable . the ordered state thus consists of two symmetric sublattices , and we can define an order parameter @xmath7 as half of the difference between the @xmath9-concentration in each sublattice . in this way , @xmath11 when all the @xmath9 atoms are on the one sublattice and @xmath12 when they are on the other . at high temperature , a disordered state @xmath13 arises . it is now easy to realize that for asymmetric initial conditions ( i.e. an unequal amount of @xmath9 and @xmath10 atoms ) the system will not be able to completely order ( strictly speaking , this is only true at low - enough temperature ) . hence , as opposed to model  a , the disordered phase can coexist with the two ordered phases . on a typical equilibrium phase diagram in the concentration - temperature ( @xmath6-@xmath14 ) plane ( fig . [ f1 ] ) , one can thus distinguish , apart from a disordered region and an ordered region , a coexistence region . the dashed line separating the ordered and disordered regions marks a second - order phase transition . in the spinodal region inside the coexistence region ( dotted line ) , the three phases are thermodynamically unstable . models have been proposed to account for various aspects of the morphology and of the kinetics of the experimental systems ( see for instance @xcite and references therein ) . from the more theoretical point of view of universality issues , the situation is not quite satisfactory . for instance , the critical exponents , and in particular the dynamic critical exponent , are still debated @xcite . a renormalization group analysis turns out to be more delicate than in the case of model  a @xcite . our goal here is to clarify the a priori simpler problem of domain growth below criticality , when the system is quenched down from a high - temperature state . notable but partial results , somewhat scattered in the literature , have been obtained in the past . for quenches into the spinodal region with droplet morphology ( quench q2 of fig . [ f1 ] ) san miguel et al . @xcite have predicted the model  b exponent @xmath0 . numerical simulations in the context of a oono - puri `` cell model '' have been found to be consistent with this prediction@xcite . on the other hand , elder et al . @xcite have predicted @xmath1 for quenches above the tricritical temperature , i.e.  in the ordered region ( quench q4 ) . to the best of our knowledge , this has not been verified numerically . our goal here is to give a complete picture of ( non - critical ) domain growth in model  c , considering , within a single system introduced in section  [ s2 ] , all four possible types of quenches illustrated in fig . this is done in section  [ s3 ] . in section  [ s4 ] , in the sake of comprehensiveness , we come back to the two following unsettled issues discussed recently in works about model  c systems . the microcanonical @xmath2 model @xcite , is a type of model  c since the order parameter is coupled to the ( conserved ) energy . zheng has suggested in a recent paper @xcite that domain growth is characterized by a non - trivial value of @xmath5 ( @xmath15 ) . a more careful study by us showed that the data are in fact consistent with the model a exponent @xmath1 @xcite . here we detail to which phase of model  c the microcanonical @xmath2 model belongs . the morphology of domains and the related `` wetting '' issues have also been a point of contention in the past . in experiments , it has been observed that neighboring ordered domains do not merge @xcite . a possible explanation proposed in @xcite is that the domains are different variants of the same ordered structure . the simulations of @xcite seem to indicate that ordered domains do not join but `` stay separated by narrow channels of the disordered phase '' : the antiphase boundaries appear to be wetted by the disorder phase . but somoza and sagui @xcite have found on the contrary that inside the coexistence region the two ordered phases may be in direct contact . we revisit their work and resolve the controversy . a summary of our results is given in section  [ s5 ] . we choose one of the simplest versions of model c which can be written as follows : @xmath16 here @xmath17 and @xmath18 are kinetic coefficients , @xmath19 and @xmath20 represent thermal noise and @xmath21 $ ] is the ginzburg - landau free energy functional which takes generally the following form : @xmath22 = \!\int\ ! d^n r \left [ f(c,\psi ) + \frac{k_c}{2 } ( \nabla c)^2 + \frac{k_{\psi}}{2 } ( \nabla \psi)^2 \right],\ ] ] where @xmath23 and @xmath24 are diffusion constants . the function @xmath25 should satisfy a few constraints . firstly it has to be symmetric in @xmath7 : @xmath26 . it should also allow for the coexistence of a disordered phase @xmath27 with the ordered phases @xmath28 @xcite . these points correspond to minima in the free energy landscape . a possible choice is @xcite @xmath29 where @xmath30 governs the coupling between the two fields . the minima are then @xmath31 corresponding to the disordered phase and @xmath32 corresponding to the ordered phase . it is instructive to analyze the model in terms of its uniform fixed points @xmath33 . they lie on the curves where @xmath34 which are the @xmath6-axis @xmath35 and the polynomial @xmath36 . their stability is determined by the signs of @xmath37 and @xmath38 @xcite . fixed points on the @xmath6-axis change stability at the so - called ordering spinodal @xmath39 : they are stable for @xmath40 . the point on the second curve where stability changes is the conditional spinodal @xmath41 . stability occurs when @xmath42 . consider now the evolution of a two - dimensional system starting from random initial conditions disordered around @xmath13 with some fixed concentration @xmath43 . we briefly present the expected follow - up of quenches at various @xmath43 values before describing our corresponding numerical results in the next section .     for @xmath44 ( disordered phase ) , the system evolves towards the stable point @xmath45 . for @xmath46 , the three phases @xmath47 and @xmath48 can in principle coexist .     for @xmath49 ( quench q1 ) , however , @xmath45 is still stable and it will be reached if initial fluctuations are small enough .     inside the region @xmath50 ( quench q2 ) , the system necessarily undergoes spinodal decomposition . for @xmath51 ( quench q3 ) , even though the ordered fixed points are now stable , nothing is changed since the initial conditions ( chosen around @xmath13 ) lead to the formation of disordered domains .     for initial concentrations @xmath52 ( quench q4 ) , the system will not phase separate but will order , so we identify this region with the ordering regime . in order to investigate the late stages of domain growth , eqs . ( [ c1]-[c2 ] ) are now studied numerically . we perform quenches from the high - temperature disordered phase to @xmath53 . ( the noise term in eqs . ( [ c1]-[c2 ] ) can be set equal to zero , since the growth exponent is expected to be determined by the `` @xmath53 fixed point '' @xcite . ) in practice , initial conditions are randomly distributed around @xmath45 . more specifically , we choose @xmath54 $ ] and @xmath55 $ ] .    equations ( [ c1]-[c2 ] ) are numerically solved on a two - dimensional grid . for the time integration we use euler s method and we approximate the laplacian by : @xmath56\ ] ] where @xmath57 and @xmath58 are the sets of nearest and next - nearest neighbors of site @xmath59 .    in most regimes studied below , @xmath60 and @xmath61 allow for a smooth representation of both fields . without loss of generality , we set : @xmath62 and @xmath63 . we record the typical domain size @xmath3 of both the order parameter and the concentration fields , as determined by the mid - height value of @xmath64 , the normalized two - point correlation function calculated for simplicity along the principal axes of the lattice using the the reduced `` spin '' variables @xmath65 and @xmath66 : @xmath67      quenches within the disordered regime being uninteresting , we first discuss q1 quenches , i.e. quenches into the coexistence region , but left of the spinodal ( @xmath68 ) . here the system possesses a stable fixed point @xmath45 but will be able to locally reach the ordered fixed points if initial fluctuations are large enough . the precise dependence on initial conditions is in fact a non - trivial problem that we have not studied systematically . this shows that the spinodal lines do not have a strict meaning in the presence of fluctuations , a well - known fact ( see for instance @xcite ) . when decomposition occurs , the ordered phase is the minority phase , and , similarly to what is observed in model  b , droplets of either ordered phases form . a similar configuration is shown in fig . the growth of ordered droplets is constrained by the conservation law . the smallest droplet disintegrates to the benefit of the larger ones . we thus expect a lifshitz - slyozov growth law @xmath69 . in our simulations , this behavior is not easily observed , due to the presence of a long transient during which the ordered domains `` nucleate '' out of the metastable state . in particular , the growth of the typical lengthscales @xmath70 does not show the expected scaling although the trend at large times is good . in this regime , though , the droplet morphology of the ordred domains can be exploited to provide a more direct measurement of growth . in fig . [ f3 ] , we show the evolution of the total number of the ordered droplet , s @xmath71 , and of the average droplet volume @xmath72 . these quantities scale respectively as @xmath73 and @xmath74 . due to a long crossover , the expected behavior is only clearly seen when plotting @xmath72 and @xmath75 against @xmath74 ( fig.[f3]b ) . the same scenario is observed for q2 quenches provided the ordered phase be the minority phase , i.e. for @xmath76 . a typical snapshot is shown in fig . we have also verified that indeed a growth exponent @xmath0 is observed . now the crossover is shorter and we see that the local slope of @xmath77 vs @xmath78 approaches @xmath79 at late times ( see fig .  [ f5]ab ) . the same conclusion is reached studying the total number of domains and their average volume ( fig . [ f5]cd ) .      due to the coupling to the non - conserved order parameter , the symmetry between high and low concentration phases , which exists in model  b , is broken . for @xmath80 the minority phase ( now the low concentration phase ) does not form droplets but is instead concentrated along lines which correspond to interfaces of the non - conserved field . in the context of alloys one speaks of `` wetting of antiphase boundaries '' . looking at the non - conserved field , we first observe model  a like coarsening which then slows down , since the interfacial regions separating ordered domains get wider ( made of the disordered phase , their volume is conserved ) . we thus again encounter the situation where the ordering is constrained by the conservation law , but with the difference that the disordered phase has a particular morphology . it is therefore not obvious that the growth exponent will be @xmath0 . the derivation of @xcite is only valid in the case of isolated droplets . however , the lifshitz - slyozov law is known to apply more generally , provided that 1 ) coarsening is driven by surface tension , 2 ) transport is by diffusion through the bulk and 3 ) the length scale that describes the coarsening process is the only relevant length scale in the system @xcite . it is easy to convince oneself that the first two conditions are fulfilled . at first sight , however , it appears that there are two lengthscales : one associated with the thickness of the disordered domains , @xmath81 ( equivalently of the interfaces between ordered domains ) , the other with their curvature ( equivalent to the lengthscale of the ordered domains ) , @xmath3 . one has to realize that the total interface length @xmath82 is inversely proportional to @xmath3 . since the volume occupied by the interface is conserved we also have @xmath83 , and therefore @xmath84 . one thus expects again a growth exponent @xmath0 . the above discussion not only applies to q2 quenches ( @xmath85 ) , but also to q3 quenches ( @xmath86 ) . a uniform configuration @xmath33 is then stable , but for symmetric initial conditions there are always interfaces where @xmath13 and where it is thus more favorable to have @xmath87 . in the `` ordered '' domains @xmath6 will then increase such that we will end up with all the three phases . all this is confirmed by our numerical results . in fig . [ f6 ] we show snapshots of order parameter and concentration at late times for @xmath88 and @xmath89 . in fig . [ f7 ] we plot , for the same concentrations , the typical lengthscale of ordered domains and disordered domains . they behave similarly . we can also identify two transient effects . initially , rapid growth takes place due to the ordering process . after that , when the antiphase bounderies are wetted , the growth will actually be slower than @xmath90 . since the disordered phase has a large surface compared to its volume , surface diffusion will dominate at short times , and this is known to lead to a @xmath91 growth law @xcite .      for @xmath92 ( q4 quenches ) no spinodal decomposition occurs . if initially a domain of the disordered phase exists , it disappears . although the concentration is not completely uniform but somewhat lower at the interfaces of the ordered domains , as seen in fig . [ f8 ] , this does not have any influence on the coarsening of the order parameter . thus , the model  a exponent @xmath1 should be observed for @xmath93 . again this is confirmed in our simulations . we have measured the typical domain size for the order parameter around @xmath94 . as can be seen in fig . [ f8 ] a growth exponent @xmath0 is still observed for @xmath43 slightly below @xmath94 , at @xmath95 . at @xmath96 , growth is clearly faster but the local exponent ( inset of fig . [ f8]c ) does not reach @xmath97 in accessible times . the behavior , though , is consistent with @xmath1 . this is most clearly observed plotting @xmath3 againt @xmath98 ( fig . increasing @xmath43 beyond @xmath94 , the crossover is now short enough and the @xmath1-behavior is apparent even in a log - log plot ( fig . the microcanonical @xmath2 model has lately received renewed attention largely because it offers an interesting bridge between statistical mechanics and deterministic dynamics @xcite . its equations of motion derive from the well known lattice @xmath2 hamiltonian . they can be written : @xmath99 where the sum is over the 4 nearest neighbors of site @xmath59 on a square lattice .    as stated before , this model is believed to be in the model  c universality class since the order parameter @xmath100 is coupled to the conserved energy . @xcite indeed , looking at snapshots of the order parameter and the local energy , defined as : @xmath101 one observes that @xmath102 is higher at the interfaces of ordered domains ( fig . [ fig - phi4]a ) as observed in our model  c around the coexistence concentration @xmath94 . given the results described above , the question of the precise relationship between the microcanonical @xmath2 model and model  c now boils down to whether we are in the interconnected ( @xmath103 ) or the ordered regime ( @xmath104 ) of model  c , i.e. whether the interfaces ( equivalently the disordered domains ) will thicken or not .    our observation that , in fact , @xmath1 @xcite indicates that we are in the ordered regime this implies that the conserved field does not undergo spinodal decomposition . the snapshots of the order parameter and the local energy taken during coarsening ( fig . [ fig - phi4 ] ) reveal that the width of the `` antiphase boundaries '' does not grow . instead , energy flows into the bulk of domains in the form of kinetic energy , as observed in @xcite . the increase of kinetic energy compensates the energy lost due to the decrease of the total interface length . we believe this constitutes further support to our result @xmath1 , at least for the parameter values studied in @xcite and @xcite . whether a coexistence regime in the @xmath2-model exists for other parameter values remains , strictly speaking , an open question . in fact , since the system is isolated , the total energy and the `` temperature '' can not be varied independently . upon increasing the energy , the temperature , which can be identified with the kinetic energy , will also increase . we therefore only have access to a line in the @xmath105-plane of fig . ( note that increasing the energy corresponds to decreasing the concentration . ) investigations of the critical behavior of the model , where it is a priori important whether one enters the disordered region from the coexistence region or from the ordered region , are under way @xcite . a few years ago , somoza and sagui have investigated the effect of interfacial properties on the morphology of domains in a model  c system @xcite . they came to the conclusion that apart from a `` complete wetting '' regime where two `` ordered '' domains ( @xmath106 ) are always separated by a disordered layer ( @xmath13 ) , there exist also a `` partial wetting '' and even a `` partial drying '' regime where ordered domains of opposite sign can be in direct contact . these regimes were observed by varying the diffusion constants @xmath23 and @xmath24 keeping @xmath107 and @xmath108 fixed . thus diffusion of one quantity was enhanced with respect to the other . we first repeated the simulations of somoza and sagui using @xmath109 . the ordered domains are in direct contact but the corresponding interfaces are not smooth , an indication that the equations are not well resolved . next , we performed the same quench but at a higher numerical resolution ( @xmath110 ) . as shown in fig . [ f9]b , no direct contact between ordered domains can be observed . the argument put forward in @xcite is that in the partial drying regime the surface tension is lower between the two ordered phases than between the ordered and disordered phases . this surface tension was measured by calculating the free energy of a numerically obtained interface profile @xcite . it must be noted , however , that this procedure is only correct when the obtained profile is the equilibrium profile . an order - order interface for instance will desintegrate into two disorder - order interfaces when the surface tension of the latter is lower . the speed at which this happens depends on the diffusion constants , since some amount of the conserved quantity has to diffuse away from the interface . one could thus have the impression that an order - order interface is stable , when in fact it is very slowly disintegrating . we have provided a comprehensive account of domain growth in two - dimensional model  c systems , unifying and sometimes correcting partial results present in the literature . quenches into the coexistence region ( @xmath111 ) all lead to @xmath0 domain growth governed by the conserved field , independent of the morphology of domains , but subjected to the strength of initial fluctuations when @xmath112 . quenches into the ordered region are asymptotically dominated by the ordering process , leading to @xmath1 growth after a crossover from slower @xmath0 coarsening . we also clarified the status of the microcanonical @xmath2 model and showed that in this case sub - critical domain growth correspond to quenches into the ordered region of model  c.    the picture that thus emerges is that domain growth in model  c systems below criticality is governed by the growth exponent of either model  a or model  b. at criticality , on the other hand , one may expect exponents different from model  a or b. the natural question that then arises is how the exponents depend on on the initial concentration . another issue open for further investigation is the behavior of the so - called persistence probability @xmath113 , defined as the probability for a spin to have remained in its initial phase from @xmath114 up to time @xmath115 ( see @xcite for a review ) . in the case of model a and b this quantity decays algebraically with an exponent @xmath116 , estimated to be @xmath117 @xcite and @xmath118 @xcite . the universality of these exponents is an ongoing debate . measurements of the persistence exponent in model  c , especially in the interconnected regime where one would expect the model  b value , might help to settle this issue .
we present a comprehensive picture of ( non - critical ) domain growth in model  c systems where a non - conserved scalar order parameter is coupled to a conserved concentration field . for quenches into the region 
 where the ordered and disordered phases coexist , we confirm earlier partial numerical results and find a growth exponent @xmath0 . for quenches into the ordered region , we confirm the theoretical prediction @xmath1 . 
 finally we discuss the implications of our results for domain growth in the microcanonical @xmath2-model and we offer some criticism of the work of somoza and sagui on the morphology and wetting properties of domains .    2