Landau Model

Continuous, or the second-order, phase transitions can be very spectacular, because they give rise to a diverging correlation length and hence to behavior known as critical phenomena. Landau theory in physics was introduced by Landau in an attempt to formulate a general theory of the second-order phase transitions. He was motivated to suggest that the free energy of any system should obey two conditions: the free energy is analytic, and the free energy obeys the symmetry of the Hamiltonian.

A generic system to discuss the transitions is the magnet. Uniaxial fer-romagnet is a simplest example of phase transitions. As we know, G = U−T S−HmagM has to be minimized at constants P and T . The big question is a minimum with respect to variables. When symmetry is broken, one needs to introduce one or more extra variables to describe the state of the system.

In the ferromagnic phase, magnetization M vanishes at certain temperature Tc at Hmag = 0. Below Tc a spontaneous M is observed, its direction can be “up” and “down” with equal likelihood. The nearest neighbors interact in a way favorable to point in the same direction. At Hmag = 0, G consists of two terms of U and −T S. At low T, U is more important and the system is a ferromagnet. At high T, G is minimized by disordered state in which S is large. It is the fight between the order and disorder that makes the critical state specifically interesting. We can denote the variables by M ; in a generic case, it is called the order parameter θ. In some cases it is difficult to specify θ and in general it can be a scalar, a vector, a tensor, etc. A basic element of Landau’s theory for continuous phase transitions is the presence of a θ.

θ is a measure of the degree of order in a system; the extreme values are 0 for total disorder and 1 for complete order. Landau supposes here that a given system can be described by a single θ, which should be zero at high T , usually above Tc, for disordered phase, and it is non-zero in an ordered phase. Examples include the dielectric polarization in a ferroelectric system, the fraction of superconducting electrons in a super conductor, or the fraction of neighbor-A-B bonds to total bonds in an alloy AB.

Landau’s unified theory of all the second-order phase transitions concerns itself with what is happening in the vicinity of the phase transition. In this region the magnitude of θ will be small as T → Tc. The Landau approach expands the free energy as a power series in these small parameters where the free energy is assumed to be an analytic function of θ, which leads to a phenomenological expression for G,

G(P, T, θ) = G0(P, T ) + a(P, T )θ + b(P, T )θ²+ c(P, T )θ³+ d(P, T )θ⁴+· · ·.

5.2 Landau and Ising Models for the Second-order Phase Transitions 161

First we note that G0(P, T ) can be ignored here since the origin of the energy is entirely arbitrary. The coefficients in the Landau expansion a, b, c, and d are function of P and T . We know that above Tc, θ is vanished and θ has some finite value below Tc. The minimum of the free energy below Tcshould therefore occur at θ = 0 and above Tc at θ = 0(∂G/∂θ = 0). From this we conclude immediately that a = 0 (for systems without external fields), because otherwise θ = 0 at any T . Also b in front of the quadratic term in the free energy should be positive for T > Tc (minimum at θ = 0) and negative at T < Tc (minimum of G at θ= 0). The simplest choice is

b(P, T ) = a0(T − Tc) (5.3) where a0is positive constant and Eq. (5.3) is only valid in a neighborhood of Tc. The condition that θ is finite below Tc requires d(P, T ) > 0 and that a0

and d are sufficiently large that all the interesting behavior occurs for small θ. Thus, we don’t have to worry about higher order functions. Furthermore, Landau’s idea is to forget the details of the microscopic model and consider just the symmetries. Hence, the power series of the G must only contain terms which respect the symmetry of the θ(c = 0), so that G can be expanded into G(P, T, θ) = a0(T− Tc)θ²+ dθ⁴. (5.4) To find the minimum, we set the derivative with respect to θ to zero, (∂G/∂θ)_T

= 2a0(T − Tc)θ + 4dθ³ = 0, which has the roots

θ =

⎧⎪

⎨

⎪⎩

0 T > Tc

−a0

2d (T − Tc)

_1/2

T < Tc

with a0 and d positive. θ = 0 corresponds to the minimum of the G function (Eq. (5.4)) at T > Tc where G(T ) = G0. The other root, θ = [a0(Tc− T )/(2d)]^1/2, is related to the minimum of G(T ) at T < Tc where

G(P, T ) = G0− a²0(Tc− T )²/(4d).

The variation of G(θ, T ) as a function of θ² for three representative T is shown in Fig. 5.2 (a), and θ(T ) function is shown in Fig. 5.2 (b).

Landau model describes a phase transition in which θ→ 0 as T → T^c. S =

−∂G(θ, T )/∂T = S0−a0θ²where S0=−∂G0/∂T , which is the entropy drop as the ordered phase is entered and it is continuous at the transition,

T > Tc, θ = 0, S = S0(T );

T < Tc, θ²= a0(Tc− T )

2d , S = S0(T ) +a²₀(T− Tc)

2d .

C_P = T (∂S/∂T )_P is then,

C_P =

⎧⎨

⎩

T (∂S0/∂T )_P = C_P,0 T > Tc

C_P,0+a²₀T

2d T < Tc

162 Chapter 5 Thermodynamics of Phase Transitions

Fig. 5.2 (a) Landau G function versus θ²at representative T . As T drops below Tc

the equilibrium value of θ²gradually increases, as defined by the position of Gmin. (b) Typical behavior of θ as a function of T . Below Tc, θ is finite, which vanishes at T > Tc.

At the transition there is a discontinuity in C_P given by ΔC_P = a²₀Tc/2d, in accord with the mean field theory. However, the magnitude of the disconti-nuity can be obtained in terms of the Landau parameters.

Taking the magnetization as an example, we can specify M as an order parameter. As a result, G depends on T, Hmag, M . Those are the intensive parameters characterizing the state. Landau’s theory corresponds to a mean field theory that ignores the effect of fluctuations, and hence gives incorrect predictions of critical exponents and, occasionally, fluctuations may even pre-vent the transition from being continuous. In spite of this, the application of Landau’s theory to predicting and understanding the symmetry changes at the transition point and the qualitative behavior of the system seems to be quite successful so far. The great virtue of Landau’s theory is that it makes specific predictions for what kind of non-analytic behavior one should see when the underlying free energy is analytic.

5.2.2 Ising Model [6, 7] and its Applications

Far and away the most influential model of a system capable of a phase transition is the Ising model. This was invented by Lenz in 1920 as a simple model of a ferromagnet, though we shall see that it can be interpreted as a model of other systems too. Magnetism (or electromagnetism) is one of the fundamental forces of nature. A field of magnetic force is produced by the motion of an electrically charged particle, so electric current (which con-sists of moving electrons) produces a magnetic field. In 1925 Uhlenbeck and Goudsmit hypothesized that the electron has a “spin”, and thus behaves like a small bar magnet. In an external magnetic field, the direction of the elec-tron’s magnetic field is either parallel or antiparallel to that of the external field.

In the same year Lenz suggested to his student Ising that if an interaction

5.2 Landau and Ising Models for the Second-order Phase Transitions 163

was introduced in between spins so that parallel spins in a crystalline lattice attracted one another, and antiparallel spins repelled one another, then at sufficiently low T the spins would all be aligned and the model might provide an atomic description of ferromagnetism. Thus the “Ising model” arose in which “spins”, located on the sites of a regular lattice, have one of two values, +1 and –1, and spins with spin values s_i and s_j on adjacent sites interact with an energy−J^s_is_j where J^ is a positive real number. Thus, spins with similar values interact with an energy−J^, and those with dissimilar values interact with the (higher) energy J^. The magnetization per spin of a system of N spins is defined as 

i

(s_i/N ) which thus lies between−1 and +1 and the total energy function (the “Hamiltonian”) is defined as the sum of the interaction energy, i.e.,

where Hmagbreaks the symmetry, the subscripts label lattice sites, and J_ij is defined that J_ij^ = J^ where i and j are neighbouring sites, otherwise J_ij^ = 0.

The model’s partition function can now be written as

ZIsing=

where {si} indicates that the sum should be extended over all possible as-signments of±1 to lattice sites.

Ising studied the simplest possible model consisting simply of a linear chain of spins, and showed that for this d = 1 case there is no (non-zero) Tc

(i.e., the spins become aligned only at T = 0). In 1944, Onsager solved the model for d = 2 in the absence of an externally applied magnetic field and showed that the model’s critical exponents were quite different from those predicted by Landau’s theory, which had been thought correct. An exact solution for the d = 2 model in non-zero external field has only recently appeared. Despite decades of intensive effort, we still have no exact solution for d = 3.

The Ising model can be mapped into the lattice gas, which is a simple model of density fluctuations and liquid-gas transitions. Since the kinetic en-ergy doesn’t depend on the position only on the momentum, the statistics of the positions only relate with the potential energy, and the thermodynam-ics of the gas only relies on the potential energy for each configuration of atoms. We divide the d-dimensional space occupied by the gas up into cells of just the same size as an individual atom. Each atom is obliged to occupy a single cell, and no cell may contain more than one atom. Let Ri be 0 or 1 decided by whether the cell is occupied or not. Since the gas is non-ideal, atoms attract each other and the energy of the gas is lower when atoms are

164 Chapter 5 Thermodynamics of Phase Transitions

in adjacent cells than when each lives in glorious isolation. If the attraction is only between nearest cells, the energy is reduced by −4JRiRj for each occupied neighboring pair.

The density of the atoms can be controlled by adding a chemical poten-tial, which is a multiplicative probability cost for adding one more atom. A multiplicative factor in probability can be reinterpreted as an additive term in the logarithm energy. The extra energy of a configuration with N atoms is changed by μN . The probability cost of one more atom is a factor of exp(−βμ). As a result, the energy of the lattice gas is E = −2

ij

J^RiRj− μ

i Ri. In order to show the correspondence between the lattice gas and the Ising model, we make the variable transitionRj= (s_i+ 1)/2 and obtain

E =−1

Therefore, the lattice model is isomorphic with the Ising model: “spin up”

in the Ising model corresponds to an occupied cell in the lattice model, “spin down” is related to an empty cell. In the Ising model, Hmagdepends (within constants) on μ and the coupling constant is 4J^.

β-brass is an alloy consisting of equal numbers of Cu and Zn atoms. At T

= 0 the alloy consists of two interpenetrating cubic lattices, one of Cu and one of Zn atoms, in such arrangement that each Cu atom is surrounded by eight Zn atoms, and vice versa for each Zn atom. As T is raised, more and more Cu atoms stray onto the Zn sub-lattice and vice versa, until at 739 K the division into two distinct sub-lattices breaks down altogether. Above 739 K, both sub-lattices contain equal numbers of each kind of atoms. This system can be described by the Ising model as follows:

At low T , the system is ordered because it is energetically preferable to unlike atoms to become the nearest neighbours rather than the like atoms.

Suppose that the system’s energy is lowered by an amount J^ for every bond between unlike atoms on adjacent sites, and raised by J^ for every bond between like atoms on adjacent sites. By a suitable choice of the arbitrary zero point of the energy scale, we can ensure that J^= J^.

Now we set the order parameter on the i-th site s_i to +1 if the site is occupied by a Cu atom, and to −1 if it is occupied by a Zn atom. Then the system’s energy function becomes E =−1

2

ij

J_ij^ s_is_j, which is identical with Eq. (5.5) for the Ising model’s energy function in the case of Hmag = 0.

5.2.3 Critical Exponent [8, 9]

Critical exponents describe the behavior of physical quantities near contin-uous phase transitions, which are associated with the emergence of power

5.2 Landau and Ising Models for the Second-order Phase Transitions 165

law distributions of certain physical quantities. A power law is a special kind of mathematical relationship between two quantities, which is some poly-nomial relationship that exhibits the main property of scale invariance. In physics and mathematics, scale invariance is a feature of objects or laws that do not change if length scales (or energy scales) are multiplied by a com-mon factor. The most comcom-mon power laws relate two variables and have the form of f (x) = ax^K + o(x^K), where a and K are constants, K is typically called the scaling exponent, and o(x^K) is an asymptotically small function of x^K. Given a relation of f (x) = ax^K, scaling the argument x by a con-stant factor causes only a proportionate scaling of the function itself. That is, f (cx) = a(cx)^K= c^Kf (x)∝ f(x).

Suppose f (Tr) is a function describing the behavior of a physical quantity, such as magnetization, where Tr= (T − Tc)/Tc, the reduced temperature Tr

is zero at the phase transition. As Tr→ 0, the limit Λ of ln[f(Tr)]/ ln tr, if it exists, is called the critical exponent associated with f . In this case, we write f ∼ Tr^Λ(“∼” is read as “is asymptotically equal to”). It is important to remember that this only represents the asymptotic behavior of the function f (Tr) as Tr → 0. In other words, a number of quantities show a power law behavior close to Tc, e.g. M (T )∼ (T^c− T )^βc.

Other examples of critical exponents for thermodynamic quantities are C∼ |Tc− T |^−αc,

χ = (∂M/∂Hmag)_T ∼ |Tc− T |^−γc, M (Hmag, Tc)∼ H^mag1/δc.

There was an astonishing empirical fact to explain the coincidence of the crit-ical exponents in very different phenomena, such as magnetic systems, super-fluid transition (λ transition), alloy physics · · ·. These phenomena, whereby dissimilar systems exhibit the same critical exponents, are called universal-ity. Universality is an important concept of the theory of continuous phase transition. Systems of the same dimension and with a phase transition into an ordered state with the same symmetry belong to the same universality class. They have essentially the same critical properties.

One of the successes of the modern theories of critical phenomena is in finding relations between the various critical exponents-scaling theories. Thus phase transitions in many different systems may be described by the same un-derlying scale-invariant theory. In fact, Scale invariance is a feature of phase transitions in diverse systems. The key observation is that near a phase tran-sition or critical point, fluctuations occur at all length scales. Diverse systems with the same critical exponents, which display the identical scaling behavior as they approach criticality, can be shown to share the same fundamental dy-namics. For instance, among the critical exponents for magnetic systems are αc, βc, γc and νc. They are not all independent, and it is possible to derive inequalities such as αc+ 2βc+ γc 2.

166 Chapter 5 Thermodynamics of Phase Transitions

5.3 Thermodynamics of Martensitic [10] and Bainite Transitions [11]

Martensitic phase transitions are first order, diffusionless, shear (displacive) solid state structural changes, which can be induced either by variation of T or by application of stress. It is the origin of properties such as shape memory effect, superelasticity, and high damping capacity (internal friction). The ap-plication of thermodynamics to understanding the martensite transition has been extremely productive over the last several decades in moving towards the generalization of effects in several steel systems.

During such transitions, a parent phase (the austenite phaseγA) trans-forms into a crystallographic different product phase (the martensitic phase αM) without any change of composition where γA is the high temperature phase. On cooling at a proper rate, αM starts to nucleate at the marten-site start temperature Ms. Upon further cooling, further nucleation of other γ^A and growth occur in such a way that α^M progressively invades γ^A. The transition finishes when allγ^Ahas been replaced byα^Mat the martensite fin-ish temperature Mf. Depending on the final temperature reached on cooling (T > Mf), the transition could stop before the whole system has changed into αM, i.e., a mixture ofγAandαMis obtained. On heating, the reverse transi-tion occurs starting at the austenite start temperature (As) and finishing at the austenite finish temperature (Af).

A general scheme of G function ofγA andαMand their difference ΔG is reported in Fig. 5.3 as a function of T . At high T,γA has lower G and thus is more stable thanαM.

Kaufman and Cohen, in their pioneering work, first introduced the use-ful concepts of T0 temperature and driving force, and established a ther-modynamic framework, which can be applied to martensitic transitions. As shown in Fig. 5.3, G ofγAandαMat T0are equal, or thermodynamically in (metastable) equilibrium, i.e.,

G(γA, T0) = G(αM, T0). (5.6) At any other T , there is a difference in G between αM and γA, which is a quantitative measure of the driving force for the martensitic transition, the larger in a positive sense the greater the driving force. According to the scheme in Fig. 5.3(b), for γA → αM transition, this driving force can be defined as

ΔG(γA→ αM, T ) = G(αM, T )− G(γA, T ), (5.7) whereas for the reverse transition,

ΔG(α^M→ γ^A, T ) = G(γ^A, T )− G(α^M, T ). (5.8) The experimental determination of T0depends on bracketing it between Ms

and As. The hysteresis between Msand As can be reduced by plastic defor-mation, thus closing the gap to a few degrees. However, in non-thermoelastic

5.3 Thermodynamics of Martensitic and Bainite Transitions 167

Fig. 5.3 Schematic representation of G functions of the parent (γA) and marten-sitic (αM) phases (a), ΔG function between the above two phases (b) and ΔG function where ΔG(γ^A → α^M, Ms) = ΔG(α^M → γ^A, As) (c). (Reproduced from Ref. [10] with permission of Elsevier)

alloys, as in Fe-C system, hysteresis between Ms and As can be as large as hundreds of degrees.

When the entropy difference between the two phases is constant, and ΔG(γA→ αM, Ms) = ΔG(αM→ γA, As) [Fig. 5.3(c)], the following equation holds for non-thermoelastic transitions,

T0= (Ms+ As)/2. (5.9)

When there are no magnetic contributions to G, C_P of a solid pure ele-ment or an alloy presents a smooth trend, and is approximately equal in the considered temperature range where martensitic transitions usually oc-cur. This implies that the ΔS(γA → αM) can be assumed to be constant.

On the contrary, when magnetic transitions occur, C_P shows the so-called λ shape, i.e. a sharp peak at Curie or N´eel temperature. As a consequence, the difference in C_P between the parent and martensitic phases can be large, ΔG(γA → αM, T ) is not a linear function of T and the driving forces ΔG(γA → αM, Ms) = ΔG(αM → γA, As). Eq. (5.9) is invalid. This occurs indeed very often. For instance, magnetism is present in Fe-base alloys. Even when the aforementioned hypotheses are not fulfilled, however, Equation (5.9) is commonly used as a reasonable approximation for obtaining T0 in non-thermoelastic martensitic alloys.

In thermoelastic alloys, hysteresis between Ms and As is limited, and thus bracketing is not required to estimate T0. Unfortunately, it has been

168 Chapter 5 Thermodynamics of Phase Transitions

found for some β-brass that A^s < T0, and thus the determination of T0

requires special care. The origin of this behavior is related to the accumulation of strain energy in the direct reaction (on cooling), which is sufficient to

“prematurely” start the reverse transition on heating. However, it has been recognized that for several thermoelastic transitions, it can be assumed that the elastic contribution becomes negligible at Ms and Af, i.e. for the first plate of martensite to form during the direct transition, and for the last plate to disappear during the inverse transition. Thus, T0 should lie at half way between Msand Af and Eq. (5.9) should be modified to

T0≈ (Ms+ Af)/2. (5.10)

Figure 5.4 reports G(T ) functions of equilibrium phases. For the purpose of illustration, a generic Fe-X system is considered and G curves of γ^A and α^M as a function of composition at different T are shown. For an alloy com-position x0 at a temperature T1 being higher than T0 (Fig. 5.4(a)), γA is stable, and its G(T1) is given by point A^. When T is decreased down to T0

(Fig. 5.4(b)), the G values of bothγA andαMare equal (point B). However, according to the common tangent rule, the equilibrium state of the system is a mixture of γA and αM, whose G is given by point B^. At any lower T3

(Fig. 5.4(c)), γA (point C) has the possibility to transform intoαMwithout

(Fig. 5.4(c)), γA (point C) has the possibility to transform intoαMwithout

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