A phase transition of first order occurs, when the freeenergy of a thermodynamical system F(T,V,N), with temperature T, volume V and particle number N, has two local minima. This means, that the system has two different homogeneous equilibrium states. These minimima are then necessarily divided by an energy barrier, that is, a region of concave F. This has the immediate consequence, that there is a certain region V _{I} < V < V _{II} in which a homogeneous state is energetically disadvantageous compared to a heterogeneous state, where the system’s material is divided into coexisting parts with the densities n_{I} = N∕V _{I} and n_{II} = N∕V _{II}. Realizations of such heterogeneous states are commonly observed, for example, in the liquid/vapour transition of H_{2}O or CO_{2}, where the coexistence of liquid and vapour can readily be prepared experimentally.
In order to illuminate this phenomenon, we will first determine the boundaries of the coexistence region V _{I},V _{II} and then calculate the freeenergy of the heterogeneous state.
In equilibrium, the coexistence of two phases is possible if and only if pressure p, temperature T, and chemical potential μ are equal in both phases, that is, throughout the entire system. Therefore, one can use the condition

where p_{I∕II} denotes the coexistence pressure, together with the GibbsDuhem relation G = ∑ _{j}μ_{j}, with free enthalpy G, to infer
G_{I}(T,p_{I∕II}) = G_{II}(T,p_{I∕II})  
⇔ F_{I} + p_{I∕II}V _{I} = F_{II} + p_{I∕II}V _{II}  
⇒ F_{I} − F_{II} = p_{I∕II}.  (1) 
 (2) 
we can combine the equations (1) and (2) to obtain

This result allows to find the coexistence region by a simple geometrical approach: the famous Maxwell construction. Drawing the isotherm of the homogeneous system in a pressurevolume diagram, the coexistence pressure p_{I∕II} corresponds to a horizontal line which is drawn in a way that the areas A and B in Fig. 1 are of equal size. This uniquely defines p_{I∕II}, V _{I}, and V _{II}.
The energy of the heterogeneous state is given by F = c_{I}F_{I} + c_{II}F_{II}, where F_{I} = F(T,V _{I},N) and F_{II} = F(T,V _{II},N) while c_{I} = N_{I}∕N and c_{II} = N_{II}∕N denote the fractions of molecules in the two phases. Since c_{I} + c_{II} = 1 and V = c_{I}V _{I} + c_{II}V _{II}, we can deduce the socalled lever rule:

Accordingly, we can write the freeenergy of the heterogeneous state as

which is obviously a straight line with the slope ∂F_{het}∕∂V = (F_{II} −F_{I})∕(V _{II} −V _{I}), connecting F(T,V _{I},N) and F(T,V _{II},N). Since we argued above, that the pressure p = −∂F∕∂V is constant along any isotherm in the coexistence region, we infer that the straight line F_{het} is actually tangent to F in both points F(T,V _{I},N) and F(T,V _{II},N). It is thus termed the common tangent in the literature. Due to the concavity of the freeenergy of the homogeneous state in between the two local minima, the tangent F_{het} is lying below F, making the heterogeneous states on the tangent energetically more favourable. This is shown graphically in Fig. 2.
This region in statespace in which the heterogeneous state has a lower overall freeenergy is called the binodal region, and it is the region where phase coexistence is possible. Its boundary, the binodal curve, is given by the loci of V _{I},V _{II} written as functions of another state variable such as the temperature T, yielding curves V _{I}(T),V _{II}(T). If phase coexistence is not possible for all values of T, then the two curves V _{I}(T),V _{II}(T) will meet in a critical point at a temperature T_{cr} beyond which the phases are no longer distinguishable from each other.
However, the given argument does not claim that a state within the binodal is unstable but only that it is metastable, that is, unstable to finite but not to infinitesimal perturbations. In fact, these metastable states are commonly observed, for example in form of an overheated liquid or an undercooled gas.
So, when does the homogeneous state actually become unstable? Looking at the pressure p = −∂F(T,V,N)∕∂V of the system, we note that any state with ∂p∕∂V = −∂^{2}F∕∂V ^{2} > 0, is mechanically unstable, so that the inflection points ∂^{2}F(T,V,N)∕∂V ^{2} = 0, when written as functions of another state variable, such as T, demark a boundary, the socalled spinodal curve beyond which no homogeneous state can exist. The region within this boundary, that is the region of negatively curved F(T,V,N), is called the spinodal region. In the next subsection, we will explain this instability by use of an energetic argument.
It is clear that the spinodal region is a subset of the binodal region, since in between every two minima of the freeenergy there is necessarily a region of negative curvature. By the same argument, we can deduce that spinodal and binodal meet in the critical point, because the minima smoothly approach each other for T → T_{cr}, so that minima and inflection points finally coincide.
Throughout this thesis, we will describe the surfactant monolayer in terms of its molecular number density n = N∕V . We therefore use the remainder of this subsection to formulate the freeenergy using the variable n, that is (T,V,n) := F(T,V,nV ), and write
 (3) 
denoting the system’s freeenergy density by f(T,n).
The pressure of the system is then given by
p = −  = − −  
= − +  
= − + nμ, 
 (4) 
In order to determine the mechanical stability of homogeneous states, we have to calculate
=  −  
= − − 2  
−. 

yielding

We conclude that, no matter if we look at the freeenergy in terms of variable volume V or variable density n, the stability is always determined by the second derivative of the corresponding freeenergy function.
The conditon of mechanical stability, ∂p(T,V,N)∕∂V < 0, can be substantiated by a simple energetical argument which, despite its illustrativeness, is  at least to the author’s awareness  never given in the literature.
A homogeneous state is unstable when even an infinitesimal perturbation leads to a state of lower freeenergy. Let us assume, that an initially homogeneous system of density n_{0} is perturbed with a perturbation of amplitude dn, so that a volume portion V _{+} of the fixed total volume V _{0} now has the density n_{0} + dn with infinitesimal dn. Material conservation demands, that another portion V _{0} − V _{+} is perturbed at the same time so that it now has the density n_{0} − V _{+}∕(V _{0} − V _{+})dn. Then the freeenergy difference between the perturbed and the initial state is given by
ΔF  = V _{+}f(n_{0} + dn) + (V _{0} − V _{+})f− V _{0}f(n_{0})  
= (V _{+} − V _{0})f(n_{0}) + V _{+}_{n 0}dn + V _{+}dn^{2}  
+ (V _{0} − V _{+})  
+ . 

Since V _{+} is by definition positive and less than V _{0}, we conclude that the sign of ΔF for infinitesimal dn is determined solely by the curvature of the freeenergy density in the inital state n_{0}. Note, that the above argument holds true also when we assume the perturbation to be limited to a subvolume V ′⊂ V _{0}. This means, that a local infinitesimal fluctuation around n_{0} will reduce the freeenergy of the system if and only if ∂^{2}f(n_{0})∕∂n^{2} < 0. Thus, we have reproduced the stability conditon given in the previus section by explicitly referring to the change of the system’s freeenergy.
So far we have only considered equilibrium states and investigated their stability. The actual dynamics of a system within the spinodal region is governed by the CahnHilliard equation [CH58], which describes the time evolution of the spatially varying density ρ(x,t) of the considered thermodynamical system. It can be written as a conservation law

with the flux
 (5) 
where α denotes a mobility factor. In the simple case of constant α, the CahnHilliard equation takes the form
 (6) 
It is important to recognize, that the CahnHilliard equation exhibits potential dynamics. This means that an isolated system approaches an equilibrium state, which corresponds to a local minimum of the freeenergy functional [ρ]. In the terminology of nonlinear dynamics this means, that [ρ] is the Lyapunov functional of the system.
This holds true for arbitrary positive definite mobility factors. To prove this, one has to consider how [ρ] changes with time. A short calculation yields
∂_{t}  = ∫ V d^{D}x∂_{t}ρ(x)  insert (6)  
= ∫ V d^{D}x∇⋅  int. by parts  
= ∫ V d^{D}x∇−∫ V d^{D}xα^{2}  Gauss’ law  
= ∫ ∂V d^{(D−1)}xα∇ −∫ V d^{D}xα^{2}. 
 (7) 
that is, either ρ is already in an equilibrium state or it is evolving towards one.
Cahn and Hilliard proposed the following freeenergy functional for a general isotropic system of nonuniform composition [CH58]:
 (8) 
The idea behind this model is simple: To molecules within a subvolume dV of the system it makes little difference whether they are part of a homogeneous system or of an inhomogeneous system with very small density gradients. Thus, as a zeroth approximation, the system’s overall freeenergy simply equals the sum of the freeenergies of each infinitesimal subvolume, f_{hom}(ρ())dV . One can expand the freeenergy density in powers of ∇ρ, keeping only the lowest orders:

The tensors κ^{(1,2)} and the vector L reflect the symmetry properties of the considered material. For isotropic media, one obtains L = 0 and κ^{(1)} = κ_{1}I, κ^{(2)} = κ_{2}I, yielding the freeenergy density of eq. (8), where κ = −dκ_{1}∕dρ + κ_{2}∕2.
Inserting the freeenergy functional (8) into the Cahn Hilliard equation (6), we obtain
 (9) 
From our considerations in section 1 we already know qualitatively what to expect from solutions to this equation: Homogeneous system outside of the binodal region are absolutely stable, within the binodal region they are metastable, and in the spinodal region they are unstable. Upon instability, the homogeneous state decomposes into coexisting domains of different densities ρ_{I}, ρ_{II}.
Spatially homogeneous fields ρ = = const. are always stationary solutions of the CahnHilliard equation. The linear stability of these solutions is readily obtained by looking at small perturbations ζ(x,t). Inserting ρ(x,t) = + ζ(x,t) into eq. (9), and keeping only the linear terms, we find that the time evolution of the perturbation is governed by

Using the ansatz ζ ∼ exp(λt + ik ⋅ x), one finds the dispersion relation of the perturbation, which only depends on the absolute value of the wavevector k = k, since only even powers of ∇ are applied to ζ:
 (10) 
From this expression, it is obvious, that the stability of a homogeneous solution is determined by the curvature of the freeenergy at ρ = . Physically speaking, a solution is stable, as long as it is chosen from outside the spinodal region, which is defined as the region of concave f_{hom}. Dispersion relations for three different in a system with a freeenergy of symmetric doublewell shape are shown in Figure 3. Otherwise, there is finite band of modes k with 0 < k < which are amplified with λ(k) > 0. The wavenumber k_{max}, which is maximally amplified, can be calculated from eq. (10) as

The corresponding maximal amplification is given by

With k_{max} and λ_{max} we have obtained an estimate of the inverse time and length scales of the early stages of spinodal decomposition.
However, after a short time, the nonlinearity of eq. 9 strikes and leads to a different behaviour known as coarsening of the domains which are formed during the initial pattern formation. During this process, large domains grow larger while small domains grow smaller and finally disappear. Two mechanisms can be distinguished: Firstly, two domains can directly merge into a single larger one, thereby minimizing thier overall interfacial energy. Secondly, material diffuses from the surface of small domains towards the larger ones, that is, the domains are absorbed without direct contact to other domains. This phenomenon is known as Ostwaldripening.
Figure 4 visualizes the dynamics of this process in a spacetime plot. The domain shape depends on the amount of available material. The phase with the smaller overall volume will form spherical domains in order to diminish the interfacial energy. If, however, both phases have roughly the same total volume, one will rather observe labyrinthine structures. The coarsening process will continue, until finally only one domain of each phase remains, divided by the smallest possible interface.