PHASE TRANSITIONS AT INTERFACES

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PHASE TRANSITIONS AT INTERFACES

S. Dietrich

To cite this version:

S. Dietrich. PHASE TRANSITIONS AT INTERFACES. Journal de Physique Colloques, 1989, 50

(C7), pp.C7-233-C7-244. �10.1051/jphyscol:1989725�. �jpa-00229889�

COLLOQUE DE PHYSIQUE

Colloque C7, suppl6ment au nOIO, Tome 50, octobre 1989

PHASE TRANSITIONS AT INTERFACES

S. DIETRICH

Institut fiir Physik, Johannes Gutenberg, Universitat Mainz, Postfash 3980, 0-6500 Mainz 1, F.R.G.

U S U ~ ~ . -Une i n t e r f a c e e n t r e deux p h a s e s peut p r d s e n t e r d e s changements e s s e n t i e l s dans s a s t r u c t u r e s i l ' o n s l a p p r o c h e d e l a t r a n s i t i o n d e p h a s e t r i d i m e n s i o n n e l l e d e l ' u n deux. Des phdnomenes c r i t i q u e s de m o u i l l a g e , cornme l a f u s i o n d e s u r f a c e e t l e d d s o r d r e i n d u i t p a r l a s u r f a c e , peuvent s e p r d s e n t e r . La d i f f r a c t i o n d e s r a y o n s X e t d e s n e u t r o n s 1 p e t i t a n g l e d ' i n c i d e n c e e s t u n e mdthode p a r t i c u l i e r e m e n t a p p r o p r i g e pour o b t e n i r d e s i n f o r m a t i o n s p r g c i s e s s u r c e t t e c l a s s e d e phdnomSnes c r i t i q u e s t r i d i m e n s i o n n e l s p r d s e n t a z t c e r t a i n s a s p e c t s b i d i m e n s i o n n e l s .

Abstract.

-

The interface between two phases may exhibit significant structural changes if one of them comes close to a phase transition in its bulk. Surface critical phenomena, critical adsorption, and wetting phenomena, like e.g. surface melting or surface induced disorder, can occur. The grazing incidence of X rays and neutrons is particularly well suited in order to obtain precise informations about these kind of critical phenomena which link the space dimensions two and three.

I. INTRODUCTION

If two coexisting phases a and y of condensed matter meet each other, a spatial transition region is formed where the properties of the system vary smoothly between their corresponding bulk values in the a and 7 phase, respectively. Normally the width of this transition region is determined by the sum of the correlation lengths

ti,

^{i = a ; y}

,

in the two bulk phases. Over a wide range of temperature and pressure

4

is of the order of a few

A.

(This holds for small particles; the case of macromolecules will be mentioned in Sects. 11.2, IV.) Therefore such interfaces are rather close realizations of two-dimensional objects.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989725

The question arises under which circumstances the interfacial structures can become much broader and whether they can develop even into full three-dimensional objects.

There are four mechanisms which can lead to such transformations. The first one is induced by those strong cooperative phenomena which occur close to second-order phase transitions in the bulk phase diagram. In this case and therefore also the thickness of the interface diverges. These modifications of the interfacial structure due to a continuous bulk transition are called surface critical phenomena. The second mechanism arises if upon changing temperature or pressure a third bulk phase ,O is becoming thermodynamically stable compared with a and 7 but still remains separated from them by a jrst-order bulk phase transition. In this case the L I - ~ interface can undergo a phase transition of its own by forming a macroscopicly thick /3-film at the interface position: a1 7 + 01/31 7 As a consequence these so-called wetting phenomena split the a-y interface into the a-,O and the P-7 interface by the intrusion of the ,O phase. The growth of this ,&like film from a microscopicly small value of its thickness towards the formation of a macroscopic bulk phase represents a particular clear example for physical phenomena linking the space dimensions two and three. A remarkable feature of these wetting phenomena is that the thickness of the interfacial structure grows without limit although all three relevant correlation lengths ti, i = a,@, 7 , remain microscopicly small: all participating bulk phases are separated from each other by first-order phase transitions. This must be contrasted with the aforementioned surface critical phenomena where the transformation of the interfacial structure is driven by the diverging correlation length in that one of the two coexisting phases, which undergoes a second-order phase transition.

There are two additional phenomena which lead to broad interfacial structures: the presence of massless Goldstone modes or of long-range atomic forces. As an example for the first case consider the spontaneous magnetization of a semi-infinite Heisenberg or XY ferromagnet at low temperatures. Due to spinwaves it approaches its bulk value at z =

+

^m

not exponentially but z-l, where z measures the distance from the interface position at z = 0 / I / . This order parameter profile $(z) is a pronounced example for a mesoscopic interfacial structure: the bulk value is approached so slowly that due to the divergence of

w

the excess quantity J dz [$(z)

-

$(a)] the interface has no well-defined thickness. (Such

0

mesoscopic interfaces can also be formed by surface critical phenomena; there they are called critical adsorption phenomena. I shall discuss them in Sect. II.2.)

The effect of long-range atomic forces can be demonstrated by considering the gas-liquid interface of a fluid whose particles interact via van der Waals forces which decay

r4 for large separations r between the particles. As a consequence the density profile attains the bulk gas and bulk liquid density not exponentially but with so-called van der Waals tails 121. In this case

-

and in contrast to mesoscopic interfaces -the zeroth and the first moment of the density distribution do exist. But the interfacial structure is so broad that the second moment and all higher moments do not exist! The last two examples show that large thicknesses of interfaces do not necessarily require phase transitions in the bulk.

In the last decade a considerable amount of research has been devoted to study the local critical behavior at interfaces under the condition that the corresponding bulk approaches a second-order phase transition. Since these aforts are well documented and reviewed /3,4/ I only recall some of the most important theoretical results.

11.1 Free surfaces

Following the general framework of Sect. I in this subsection I consider the case where ^(Y is the vacuum and 7 is a system undergoing a continuous phase transition. This may be either a ferromagnet close to its Curie temperature Tc or a binary alloy close to its order-disorder transition. For simplicity in the following I adopt the magnetic language in which the magnetization plays the role of the order parameter. In the bulk the magnetization vanishes as M

4

for r = IT-TcI /Tc

-

⁰

^,

the susceptibility diverges as

x

r

-

and at T, the two-point correlation function decays as G

- ^1;

^-

^7'

^for

+ ^-i

I

x - x'

I

^{+ CO.} These singularities are induced by the divergence of the correlation length

5

r -U

. ^p

^{0.32, 7 2 1.24, r] N} 0.03, and u N 0.62 are the standard bulk critical exponents which due to universality depend only on the space dimension d and on symmetry. (Here and in the following I quote their values for d = 3 and for the Ising universality class.) Since both dimension and symmetry are modified close to the interface one can expect that there the critical exponents will differ significantly from their bulk values. Indeed one finds that in a surface layer of the thickness

5

given by the bulk correlation length the critical properties are changed in the following way:

x1

is the response of the surface magnetization M, on a bulk magnetic field and

x,,

its response on a surface magnetic field. In this context surface always means a layer of thickness (. Therefore upon approaching Tc the effect of the surface propagates into the bulk without limits! The magnetization, which depends on both z and T, takes on a scaling form so that the corresponding scaling function f mediates smoothly between the bulk and the surface behavior:

with

A, A , and B are amplitudes. In the semi-infinite geometry as considered here the two-point correlation function does not only depend on the distance between the two points

2

^{= , (;} z > 0) and

2~

⁼

(?.,

z.

>

0) but also on their relative distances z and z' from the surface. At Tc one finds

l $ - ~ . ~ - ( ~ - ~ + q " ) ; ^Z, Z' fixed,

I?-;/(

^{+ m}

G N

I

1 2 - 2 / 1 - ( ~ - ~ + ~ ) ; z f i ~ e d , ~ ' + m

' _'

_-(d-2+7) ; z , z ' ,

I;-;fl

^'co,

I x - x / I

with

Note that q,, prevails for arbitrarily large but fixed values of z and z 8 . This means that at Tc the behavior of the whole system is surfacelike! Eqs. (2.1-7) show that the critical surface exponents are drastically different from their counterparts in the bulk: at the surface y changes sign and q increases by a factor of fifty. It turns out that these surface critical exponents - as well as scaling functions like in Eq. (2.5)

-

are universal and that they fulfill scaling laws, e.g.,

Eq. (2.8) states that the altered critical behavior at the surface is induced by the rapid decay -induced by missing neighbors - of the parallel correlations near the surface: a large value of q,, leads to a large value of

PI -

which means that the order parameter at the surface vanishes more rapidly than in the bulk

-

and to a negative value of y,, and thus to a cusplike singularity of the surface susceptibility. This situation prevails at the so-called ordinary transition where the missing neighbors are not compensated by an increased strength of the forces acting between the surface magnetic moments. In this case the phase transition at the surface occurs at the same temperature Tc as in the bulk and f(u) is a monotonicly increasing function. However, if the surface couplings are enhanced beyond a certain threshold a genuine two-dimensional surface ordering

-

the so-called surface transition

-

takes place at Ts

>

Tc, which upon a further decrease of temperature is followed by the extraordinary transition at T = Tc where the bulk orders in the presence of the ordered surface. These two cases are separated by the multicritical transition for which

/3, <

P.

It should be noted that for the ordinary transition there is one but only one

independent critical surface exponent. This means that if any surface exponent is known all other surface exponents follow from scaling laws and the knowledge of the bulk exponents.

A continuous phase transition is not only characterized by diverging spatial correlations but also by a diverging relaxation time. Close to the surface this critical slowing down leads to significant differences in the time-dependent two-point correlation function compared with its bulk behavior. I shall discuss these results in Sect. IV by illustrating how they determine the inelastic diffuse scattering intensity of neutrons under the condition of grazing incidence. It turns out that all singularities of the dynamics at the interface are governed by the dynamical bulk exponent z and the static surface critical exponents. (z N 2.01 for purely relaxational dynamics and the relaxation time diverges for zero momentum transfer T -m as in the bulk.) The absence of genuine dynamic critical surface exponents reflects the fact that the surface breaks the translational symmetry in space but not in time.

The static thermal behavior of binary alloys can be expressed in terms of that of the Ising model by switching from occupation numbers ri, which are zero for those lattice sites occupied by A atoms and one for those occupied by B atoms, to spin variables si = 2 ri - 1

= h 1. Due to the missing neighbors at the surface this transformation leads to a homogeneous surface field in the corresponding Ising model which causes an increased surface magnetization. In the binary alloy this corresponds to the phenomenon of surface segregation and it reflects the fact that one of the two components has a higher preference

for the surface than the other. If the bulk of the binary alloy undergoes an order-disorder transition, like e.g. in Fe3A1, the corresponding magnetic model is an Ising antiferromagnet. In this case the surface field, which is homogenous parallel to the surface, is not the conjugate field to the order parameter, which in this case is the staggered magnetization, and thus it is irrelevant. This means that the asymptotic critical surface behavior of a binary alloy, which undergoes an orderdsorder transition, is not changed by surface segregation.

If on the other hand the binary alloy undergoes a segregation transition the corresponding magnetic model is an Ising ferromagnet. In this case this homogeneous surface field couples to the magnetization and gives rise to a nonvanishing order parameter profile even above Tc. The same arguments hold for a gas-liquid phase transition in a one-component system. There ri = 0,l means that a given lattice site is either empty or occupied and the corresponding magnetic model is again the Ising ferromagnet. In both cases the aforementioned surface field leads to the phenomenon of critical adsorption which will be discussed in the following subsection.

11.2 Bounded Surfaces

The theoretical results presented in the previous subsection apply to free surfaces.

This means that at the surface the ordering degrees of freedom are not exposed to their conjugate field. These situations are realized for phase transitions in crystals where only internal degrees of freedom, like magnetic moments or occupation variables, participate.

However, many critical phenomena occur in fluid systems. On one side they offer the advantage of getting rid of lattice imperfections which impede the growth of J in crystals.

On the other side fluids require special care in order to counter the disturbing effects of gravity, e.g., by creating microgravity conditions by deuterating binary liquid mixtures. In any case fluids must be confined either by solid walls or by other fluid phases. In both cases the confining forces act on the density which happens to determine also the order parameter of these systems. Thus there are no free fluid surfaces. (A "free" water surface, e.g., requires its vapor as a confining fluid. Otherwise evaporation would occur and the system would be no longer in equilibrium. But even without such confinements the aforementioned surface field is always present and it is always relevant, because all fluid phase transitions occur via segregation.) In this subsection I discuss the structure of these bounded surfaces if the confined fluid undergoes a continuous phase transition.

There are three relevant experimental situations. In the first case one is interested in the density profile near the wall of a one-component fluid in the vicinity of its critical point on the gas-liquid coexistence curve. According to the general framework of the Introduction in this case the a phase is the wall and the 7 phase is given by the nearly critical fluid. In the second case the 7 phase is replaced by a binary liquid mixture close to its demixing transition into an A-rich liquid and into a B-rich liquid. Here the segregation of one of the two components at the wall leads to a critical concentration profile. Finally one can again consider a binary liquid mixture with the 7 phase as in the second case but now the a phase being the equilibrium vapor of the mixture. This situation is particular interesting because here the interface is no longer prescribed geometricly by the presence of a wall. Instead the adsorption of one component occurs at a fluid interface which is generated by the condition that the binary liquid mixture is at vapor-fluid coexistence and one is approaching its critical end point on the triple line. (The relevant literature is summarized in Sect. 1X.B of Ref. 5.)

Let us focus on the first case in order to describe this kind of interfacial structure.

The density profile displays a remarkable sequence of decay modes for increasing distances from the wall. Provided that the substrate potential of the wall is stronger than the corresponding fluid-fluid interaction (- otherwise there will be critical depletion instead of critical adsorption, see below -) the density reaches a high value in the close vicinity of the wall. Within the first few the substrate potential provides such a high local pressure that the fluid is solidlike and one observes packing effects which result in a layered structure orthogonal to the wall. These density oscillations decay exponentially and finally a monotonic decay of the profile is reached. After passing a transient regime the density profile starts to decay according to a power law a . ~ - ~ with p = ,8/v

-

^{0.52. At z}

-

^{( there}

is a crossover to an exponential decay

-

^{e*/( for z}

^>

(. At even further distances there is a second crossover to the decay

-

^{z - ~} dictated by the van der Waals forces (see Sect. I).

Finally the retardation of the van der Waals forces becomes relevant and the density profile approaches the value of the bulk gas density

-

z4. For T -t Tc ( diverges so that at Tc one is left with a pure power law decay

-

^z

-

^p. (At Tc the van der Waals forces are irrelevant and cause only corrections to the dominent decay ^N z ~ ' ~ ~ . ) Upon approaching

w

Tc the coverage J [p(z) - p dz diverges

- ^@

^{= T ~}due to the slow decay of p(z) at ^{. ~ ~}

0

2

Tc. For this reason this phenomenon is called critical adsorption. Since at Tc the coverage is infinite the density profile a t Tc is a pronounced example for a mesoscopic interfacial structure which means that there is no possibility to assign a thickness to it.

Here I want to add that of course this effect occurs also in magnetic systems if one succeeds to apply a surface magnetic field. (In addition these systems have the advantage that the perturbation by long-range forces is particularly weak and that the XY and Heisenberg universality classes are accessible.) This may be accomplished by coating the surface of the ferromagnet of interest by a microscopicly thin magnetic film whose transition temperature lies above that of the semi-infinite magnet. Thus the spontaneous magnetization of this overlayer provides the ordering surface field. This is equivalent to the situation an extraordinary transition without an additional overlayer. Therefore any magnetic system with a surface transition at T = Ts does exhibit critical adsorption at T

= Tc < Ts.

Critical adsorption is not only exhibited by simple fluids. Polymers provide another important application. Either at the container walls or at the fluid interface of the solvent the monprner concentration profile takes on the same scaling form as for'critical adsorption if the chain length of the polymers becomes increasingly large. Here, however, p = 3

-

zT1

1 1.30 below, and p = 1 - (l-vS)v-' 2 0.37 at the corresponding adsorption temperature 161. Whereas in the previous examples of mesoscopic interfaces the decay exponent /I is either a rational number or given by bulk critical exponents, the latter case is particular interesting. There p is determined by the genuine surface exponent cps 2 0.63 which is a second independent surface critical exponent characterizing the multicritical transition mentioned in the previous subsection 141.

Finally as a last example for mesoscopic interfaces I want to mention the critical adsorption of a fluid, which instead of a second-order phase transition exhibits a tricritical phase transition. In this case p =

'12

171.

Sect. 11. is devoted to the structure of interfaces whose corresponding bulk undergoes a continuous phase transition. As explained in the Introduction different interfacial structures, so-called wetting phenomena, arise in the vicinity of a first-order bulk transition.

The intrusion of a film of phase ,4 between two coexisting phase cr and y is a common feature. Nonetheless a thorough understanding of these interfacial phase transitions is only starting to emerge, because both theoretical and experimental

investigations of this subject face severe difficulties. Since the vast literature on wetting phenomena has recently been reviewed (see Ref. 5 and references therein) I refrain to iterate parts of this discussion. Here I only want to raise attention to certain phenomena which are of particular interest for experimental investigations using grazing incidence of X rays and neutrons.

In the standard situation for wetting phenomena a liquidlike film is formed on a solid substrate upon approaching from the gas side the bulk gas-liquid coexistence curve of the fluid adsorbate. Likewise a solid film may be formed on a substrate upon approaching the sublimation curve. In the general framework this corresponds to a = wall, y = gas and

fi

= liquid or solid, respectively. Similar phenomena arise in binary liquid mixtures where a

= wall, 7 ⁼ A-rich liquid and /3 = B-rich liquid. A wetting phenomenon at an intrinsic, not geometrically prescribed, interface is called interfacial wetting. A typical situation of this type can be found in binary liquid mixtures with a = vapor, y = A-rich liquid, and /J'

= B-rich liquid. Although all these phase transitions at interfaces follow the same pattern, they exhibit a rich variety of features, because different from bulk critical phenomena (see Sect. 11) wetting transitions display a very delicate sensitivity on details of atomic interactions. This is a direct consequence of the small value of the bulk correlation length at the wetting transition.

The main interest, both in theory and experiment, consists of monitoring the pressure and temperature dependence of the tickness of the wetting film. For this purpose reflectivity measurements of X rays and neutrons provide an excellent tool for resolving many conflicting experimental results obtained so far by various other techniques. In addition the analysis of the associated diffuse scattering intensity may be used to study the build-up of long-range parallel correlations at the emerging w/3 and /3-y interfaces (see Sect. 1V.B in Ref. 5).

X rays and neutrons with grazing incidence are also particularly well suited to study surface melting, which is an interfacial wetting phenomenon: the liquid phase may wet the intrinsic solid-gas interface of the same system if its triple point is approached along its own sublimation curve. (The corresponding theoretical and experimental results are summarized and discussed in Sect. VIII of Ref. 5.) The vanishing of Bragg peaks in the scattered intensity of totally reflected X rays or neutrons checks the formation of a liquidlike film whose thickness exceeds the penetration depth of the X rays or neutrons /8/.

The occurrence of surface melting is proved by the vanishing of these Bragg peaks for arbitrarily large penetration depths.

Another type of wetting phenomena occurs at the interface between the ordered phase of Cu3Au (= yphase), which at Tc undergoes a discontinuous order-disorder transition, and the vacuum (= a-phase). This interface undergoes a complete wetting transition by the disordered phase upon raising the temperature towards Tc. Consequently at the surface the order parameter vanishes continuously for T -, Tc although it disappears discontinuously in the bulk. (A comprehensive discussion of these phenomena called surface induced disorder is given in Sect. VII of Ref. 5.) Again the continuous formation of a surface film of the disordered phase can bd followed by totally reflected X rays or neutrons 191. Although wetting phenomena are interfacial phase transitions their associated thermal singularities are quite different from those occurring at surface critical phenomena (Sect.

11). Thus it is a misconception / l o / to regret in the case of Cu3Au the absence of those singularities which are predicted for surface critical phenomena.

I also want to point out that the onset of orientational order at interfaces is also a wetting phenomenon. For example the interface between vapor and the isotropic fluid of liquid crystals can be wetted by a smectic phase (See Sect. XII1.B in Ref. 5).

Finally I like to emphasize that there are interesting crosslinks between surface critical phenomena and wetting phenomena. They have been investigated and discussed by Frenzl 111,121.

rV.

CRITICAL SURFACE SCATTERING

Totally reflected X rays and neutrons create an evanescent wave of penetration depth

L

below the surface on which they impinge. For surface critical phenomena L << J so that the evanescent wave probes the correlations discussed in Sect. 11.1. In a series of papers 113-181 detailed predictions for the resulting scattering intensity have been worked out. Since they have been already summarized in Ref. 19 I quote only some of the most important results.

The superlattice Bragg peak intensity stemming from a continuous order4isorder transition like in Fe3Al vanishes

- ^PA

when Tc is approached from below. Above Tc the diffuse scattering intensity reaches a finite, maximum value at the position of this Bragg peak. As function of parallel momentum transfer p and temperature r this maximum is attained via cusplike singularities:

-

^{p-l+q'' for r = 0 and T}

-

^{for p =} 0. Therefore such an experiment yields the values of three critical surface exponents and checks the scaling laws between them. The universal scaling functions are probed by varying the

penetration depth R In the case of inelastic neutron scattering at a system with purely relaxational dynamics these cusplike singularities turn into divergences which are significantly weaker than those in the bulk:

-

^{p-up for T = 0 = w, p + 0; N T} -fJ ^{7 for p = 0}

-

= w, ^T + 0;

-

^Y f J Y f ~ r p = 0 = T , w -) 0. Y is the energy transfer, op = z

+

^{n l / u N} 1.53, u = 711

+

^{YZ N} 0.93, and a, = 1

+

y ~ l / z N 0.76.

T

These results are valid for surface critical phenomena as discussed in subsection 11.1.

In order t o probe the order parameter profile for critical adsorption reflectivity measurements are required. It turns out that both the decay exponent p and the corresponding prefactor a (see Sect. 11.2) can be extracted unambiguously from the singular behavior according to which the reflectivity coefficient R reaches 1 a t the angle a ⁼ ac of total reflection 1201: for p ^E (2,mI one has 1-R

-

E f2, for p 1 = 2 1-R

-

^E-

^,

^{and for}

1 1

jp 12). These formulas apply asymptotically for p ^E (0,2) 1-R

-

exp(-2 I(p) X /E

E

-

^{*ac + 0. X} is proportional to a and I(p) =

:I2

^{I'(lrl - 1/2)/(2 I'(/)).} Thus for an exponentially decaying order parameter profile, i.e. p = m, R exhibits a square-root singularity with a positive curvature whereas in general for all those mesoscopic interfaces discussed in Sects. I,II.2, i.e. for p < 2, R has an essential singularity with negative curvature at the angle of total reflection. Such a negative curvature of R is in accordance with recent neutron reflectivity data obtained from the critical adsorption profiles of polymer solutions 1211

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