UNIVERSITÀ DEGLI STUDI DI PISA
Facoltà di Ingegneria
Dipartimento di Ingegneria Strutturale
T
ESI DID
OTTORATOCorso di Dottorato in Ingegneria delle Strutture XVIII ciclo - anno solare di inizio 2003
Settore scientifico disciplinare Scienza delle Costruzioni - ICAR 08
M
ATERIAL AND
G
EOMETRIC
P
HASE
T
RANSITIONS IN
B
IOLOGICAL
M
EMBRANES
Giuseppe Zurlo
Tutori:
Prof. Luca Deseri
Prof. Salvatore Marzano
Prof. Roberto Paroni
Chapter 1. Introduction 4
1. Phase transitions in phospholipid bilayers. 5
Chapter 2. Preliminary Notions 9
1. Mathematical preliminaries. 10
1.1. Notation. 10
1.2. Surfaces, surface differential operators, fundamental forms. 10
1.3. Surface deformations, polar decomposition theorem. 18
1.4. Some fundamental facts from differential geometry. 20
1.5. How to reconcile with the classical notation. 21
1.6. Notes of calculus of variations 25
Chapter 3. Solution Theory 29
1. Classical solution theory of fluid mixtures. 30
1.1. Basic definitions. 30
1.2. Basic thermodynamics - classical approach. 31
1.3. Material and geometric phases. 33
1.4. Heterogeneous equilibrium - The Gibbs phase rule. 35
1.5. Gibbs-Duhem equations - The common tangent rule. 39
1.6. Ideal solution theory - Raoult’s law. 45
1.7. Non-ideal solution theory - Bragg-Williams and Flory-Huggins theories. 49
1.8. Lipid-lipid and lipid-cholesterol mixtures. 55
2. Continuum thermodynamics of solution theory with deformation. 57
2.1. Preliminary definitions & basic laws. 57
2.2. The variational problem for a non-reacting mixture. 61
Chapter 4. State of the Art 63
1. State of the art. 64
1.1. Pioneering works on the morphology of thin fluid films. 64
1.2. The model proposed by Helfrich. 65
1.3. The model proposed by Baesu & others. 65
1.4. The model proposed by J¨ulicher & Lipowsky. 66
1.5. The model proposed by Sackmann. 68
1.6. The model proposed by Taniguchi. 69
1.7. The model proposed by Ayton & others. 69
1.8. The model proposed by Chen & others. 70
1.9. The model proposed by Sens & Safran. 71
1.10. The model proposed by Komura & others. 72
2. Critical comparison among the discussed models. 76
Chapter 5. The Proposed Model 77
1. Phenomenological derivation of a free energy for binary mixtures. 78
1.1. Introduction and notations. 78
1.2. Formation and ideal mixing energies. 80
1.3. Nearest-neighbor energy (1): local chemical interactions. 80
1.4. Nearest-neighbor energy (2): local elasticity effects. 82
1.5. Medium to long range interaction energy. 85
1.6. Final form of the Helmholtz free energy density. 85
2. Appendix A - The Landau expansion of the elastic energy. 86
3. Appendix B - The elastic energy density ϕ(χ, J; T ). 88
3.1. Motivations of the choice. 88
3.2. Tuning A and f (χ; t). 88
Chapter 6. Dimension Reduction 90
1. Constitutive and kinematical assumptions. 91
1.1. Experimental grounds of the models. 91
1.2. The ansatz on the shell deformation. 94
2. Asymptotic expansion of the energy density. 98
2.1. Surface energy density of a very thin, elastic body. 98
2.2. The 3D constitutive equation of a very thin, fluid, incompressible elastic
material. 100
2.3. Calculation of the tensors B and A. 101
2.4. Surface energy density of a very thin, fluid, elastic body. 103
Chapter 7. The Equilibrium Problem 106
1. The variational problem. 107
1.1. Settings and formulation of the problem. 107
1.2. A strategy for solving the variational problem. 109
2. Purely membranal problem I: temperature driven phase transitions. 110
2.1. Introduction. 110
2.2. Lipid/lipid binary mixtures. 110
2.3. Minimization in J. 111
2.4. Minimization in χ: qualitative discussion. 113
2.5. Minimization in χ: calculations. 118
2.6. Final results: temperature driven phase separation. 121
3. Purely membranal problem II: equilibrium of a closed membrane. 124
Acknowledgements 128
Introduction
1. Phase transitions in phospholipid bilayers.
This thesis concerns the study of phase transitions in phospholipid bilayers, which are the major structural elements of all biological membranes. For this reason phos-pholipid bilayers can be considered as the basic bricks of life on earth.
Figure 1. Schematic representation of a cell membrane.
Phospholipid molecules are structurally characterized by a hydrophilic, phosphate - containing polar head, which is attracted to water, and a hydrophobic, non-polar hydrocarbon chain, which repels water. The occurrence of both these features endows phospholipid molecules in water solutions of self-organizing properties, so that they tend to organize themselves into a rich variety of structures, such as, for examples, lipid bilayers.
Figure 2. The structure of an amphiphilic molecule
Biological membranes may contain hundreds of different constituents. In order to understand the basic features of these structures, experiments are performed on sim-plified model bilayer membranes, artificially created in laboratory by self-organization in water solution of two or three different kinds of lipid molecules into closed struc-tures, called liposomes. It has been observed that artificially created membranes made of three constituents mixtures of saturated and unsaturated lipid molecules and cholesterol molecules describe quite accurately the crucial properties of biological membranes. These model bilayers are closed spheroidal structures, with a thickness
of a few nanometers, and a diameter which can range from 50nm up to tens of mi-crometers; larger liposomes are generally referred to as ”Giant Unilamellar Vesicles” (GUV).
Figure 3. Schematic representation of a liposome.
There is nowadays a strong experimental evidence that rather than being uniformly distributed on the surface, the membrane constituents undergo a phase separation phenomena, as shown by the coexistence of domains characterized by different chem-ical compositions, configurational properties of the hydrocarbon chains of the lipid molecules, and elastic properties.
Figure 4. Phase separation of ”rafts”, that is domains characterized by different
chemical compositions and mechanical properties on the membrane surface.
The elastic properties of lipid membranes play a crucial role in determining the equilibrium configurations assumed by liposomes in water solution: advanced high-resolution fluorescence imaging tools have recently revealed the existence of a rich variety of composition-domain patterns and equilibrium shapes in ternary GUVs, for various values of the liposome composition, of the external temperature and osmotic pressure. It is believed that these exotic equilibrium configurations can be explained on the basis of a still unclear coupling between the chemical and the mechanical properties of lipid bilayers.
It is commonly recognized that the phase separation phenomena on cell membranes are closely associated with cell functionality: lipid phase coexistence is expected to modulate the fundamental transport processes between the organelles contained in
Figure 5. Images experimentally obtained by Baumgart, Hess & Webb [7],
which clearly show how phase separation is strictly related to the equilibrium shapes of GUV in water solution. In red and blue respectively disordered and liquid-ordered phases. Scale bars 5µm.
the cell and its environment, and it has been shown that deviations from the natural lipid composition and domain distribution may lead to severe health problems. On the other hand, in recent years a growing interest on the properties of liposomes has been driven by their strong potential in pharmacological applications, in partic-ular for their use as deliverers of anticancer agents or drugs against cardiovascpartic-ular diseases, or for their use in bio-engineering applications, gene therapy, medical di-agnostics and much more. As well as in cell functionality, phase separation is here expected to play a crucial role since closely related to the transport of drugs across the membrane.
Both theoretical studies on human cells and industrial applications of liposomes need mathematical models on the chemo-mechanical behavior of lipid bilayers. Up to now, a huge number of partial model exists, each of which focuses on limited aspects of the whole problem: there are works mainly concerned with the chemical and structural properties of lipid molecules within the bilayer (see for example [1],[35],[43],[57],[62]), works which ignore the question of chemical composition and consider the membrane as a homogenous, elastic, two-dimensional shell under various boundary and loading conditions (see for example [6],[5],[50],[51],[52]) and models which attempt to estab-lish more or less heuristic couplings between chemical and mechanical properties (see for example [34], [32],[33],[53],[60]), but a complete model on the chemo-mechanical behavior of lipid bilayers seems nowadays still un-reached.
The aim of this thesis is to propose a new phenomenologically grounded model which, on the basis of the tools of Thermo-Chemo-Mechanics, can explain how chemistry and mechanics interact in shaping lipid bilayers. In the thesis it is developed a model which fully describes the phase separation phenomena and the observed equilibrium shapes, on the basis of elementary information on the liposome nature and com-position, pressure and temperature, ruling out arbitrary assumptions on the phase separation process. We will also stay away from heuristic constitutive couplings be-tween chemical and geometrical variables, which are usually finely tuned in order to produce the desired solutions. On the basis of the Bragg-Williams and Flory-Huggins
theories of solutions and of a procedure of dimension reduction for fluid, elastic bod-ies, justified by the fact that multilayered liposomes might be of future interest, we derive a new expression of the elastic energy density for lipid bilayers. In the final part of the work we set up the variational formulation of the equilibrium problem, and we present some partial results concerning the stress-free purely membranal case, as a first qualitative discussion on the solution of the complete problem, which is still under investigation.
The structure of this thesis can be summarized as follows: Chapter 2 collects all the needed preliminary mathematical notions regarding differential geometry and surface deformations; Chapter 3 contains a review of the solution theory, both within the classical deformation-free context, and within the Continuum Thermodynamics ap-proach, in order to give a self-contained presentation of the theories at the basis of the chemical properties of lipid bilayers; Chapter 4 contains a short review of the state of the art on the argument; in Chapter 5 we provide our new phenomenolog-ical derivation of the energy density for a lipid bilayer, on the basis of the physphenomenolog-ical evidence of non-local elastic response of the considered material, and of phenomeno-logical coupling between chemical and mechanical effects; Chapter 6 concerns a procedure of 3D-2D dimension reduction which yields a new expression of the elastic energy density for a very thin, fluid membrane; finally, in Chapter 7 the complete variational problem of the equilibrium it set up, and we present some partial results.
Preliminary Notions
1. Mathematical preliminaries.
In this section we recall some basic facts about differential geometry. Point-wise dependence will be omitted when clear from the context, and stressed just when necessary.
1.1. Notation. We denote scalars with light-face, italic letters: a, n, A, N, ...,etc.; vectors with bold-face italic letters, a, n, A, N , ...,etc.; ordinary second-order tensors with face majuscules, A, B, C, ...,etc.; superficial second-order tensors with
bold-face, sans-serif majuscules, A, B, C, ...,etc. We will refer to R3as the three dimensional
Euclidian space, endowed with a Cartesian orthonormal frame {E1, E2, E3}. We use
the summation convention with the agreement that Latin and Greek indices have the ranges {1, 2, 3} and {1, 2}, respectively.
1.2. Surfaces, surface differential operators, fundamental forms. Let Ω
be an open, bounded set in R2. Any given vector z ∈ Ω admits the representation
z = zαEα = [(z − 0) · Eα]Eα.
The scalars zα will be called the cartesian components of the position vector z . A
surface S of the 3D point space R3 is defined as follows1
S := {X ∈ R3|X = p(z ) = p(z
1, z2), ∀z ∈ Ω}
being the parametrization p a diffeomorphism from Ω to S ⊂ R3(see [13], pg.74).
A natural base2 of the tangent space at each point of S can be defined as follows
Aα = ∂p
∂zα
.
Such a basis needs not to be orthonormal, and it defines the local tangent space to S
TS := Span{A1, A2}.
On the same local tangent plane TS it is possible to define the reciprocal base
{A1, A2} such that Aα· Aβ = δαβ
which forms an alternative base for the space TS. Any vector t ∈ TS can thus be
expressed in terms of its natural or reciprocal components
TS ∋ t = t
αA
α = (t · Aα)Aα (natural components)
tαAα = (t · Aα)Aα (reciprocal components).
1For a precise definition see Do Carmo,[13] pg.73.
2We use the convention to adopt majuscules for vector fields defined on the surface S; this
convention will be useful in the following, where majuscules will be used for vectors of the reference surface and minuscules will be used for vectors defined on the current surface.
The unit normal vector at each point of the surface S can be defined in terms of natural as well reciprocal basis
N := A1× A2 |A1× A2| = A 1 × A2 |A1× A2|. Letting {A1, A2, A3} := {A1, A2, N } {A1, A2, A3} := {A1, A2, N } we have that Ai· Aj = δij.
Let us now define the following fundamental linear operators, according to the defi-nitions given in Gurtin & Murdoch [21].
Identity Operators: I ∈ Lin(R3, R3) and I
S ∈ Lin(TS, TS);
Perpendicular Projector: PS is the operator in Lin(R3, TS) that projects
vectors v ∈ R3 on the surface T
S;
Inclusion Map: IS is the operator in Lin(TS, R3) that embeds any vector
τ ∈ TS in the space R3
Consider the natural basis {A1, A2, N } and reciprocal basis {A1, A2, N } for the
space R3 and the natural basis {G
1, G2} and reciprocal basis {G1, G2} for TS. Then
it results
I = Aα⊗ Aα+ N ⊗ N = Aα⊗ Aα+ N ⊗ N ∈ Lin(R3, R3)
IS = Gα⊗ Gα = Gα⊗ Gα ∈ Lin(TS, TS)
IS = Aα⊗ Gα = Aα⊗ Gα ∈ Lin(TS, R3)
PS = Gα⊗ Aα = Gα⊗ Aα∈ Lin(R3, TS).
When clear from the context and there is not danger of misinterpretations, we will
occasionally confuse vectors Aα with Gα and Aα with Gα.
It is easy to check the validity of the following properties
(1) IT
S ≡ PS;
(2) PSPTS = ISTIS = PSIS = IS ∈ Lin(TS, TS);
(3) ISPS = I − N ⊗ N ∈ Lin(R3, R3).
Let us now introduce the concept of surface gradient of the smooth vector field defined on the surface S, namely a surface vector field
Let us consider a smooth extension ˜v of the function v to a 3D neighborhood of a point X ∈ S, so that v coincides with the restriction of ˜v to the function having S as domain, that is
v(X ) ≡ ˜v( ˜X)|X3=0 ∀X ∈ S
being
˜
X = X + X3A3
with X ∈ S and X3 = ( ˜X − 0) · A3 the cartesian component of the position vector
˜
X in direction A3 ⊥ Aα. The three-dimensional gradient of ˜v reads as usual
∇˜v.
The surface gradient of ˜v at any point of S is defined restricting the domain of
∇˜v to the local tangent space of S at every point X ∈ S, that is
∇Sv˜ := (∇˜v)IS = (∇˜v)[Aα] ⊗ Gα ∈ Lin(TS, R3).
Since actually the surface vector field v and its extension ˜v coincide when evaluated
on the surface S, we can define the gradient of v as the surface gradient of its extension ˜
v, that is
∇Sv := (∇˜v)IS.
Worth noting, ∇Sv is a 3 × 2 matrix. In particular, on the basis of property (4)
enunciated in the previous page
I = ISPS+ N ⊗ N ,
we can recast ∇˜v when evaluated at points of the surface S as
∇˜v = (∇˜v)I = ∇˜v[ISPS+ N ⊗ N ] = (∇Sv)PS+
∂ ˜v
∂X3 ⊗ N
In similar fashion it is possible to show that, letting ϕ a scalar surface field defined on
a surface S ⊂ R3, and ˜ϕ its extension to a three-dimensional neighborhood of such
functions, the following definition is well posed
∇Sϕ := PS(∇ ˜ϕ) ∈ TS.
From now on, we will drop the notation˜in order to distinguish between the surface field and its smooth extension to the space.
Let now p : Ω ⊂ R2 → S ⊂ R3 be a parametrization of the surface S, then the
gradient of p can be unambiguously defined as
P:= ∇p
and it is clear that
∇p ≡ (∇p)IS =: ∇Ωp,
hence in general P ∈ Lin(Ω, R3). Nevertheless, it is evident that
as it is easy to check considering that
P= ∇p = (∇p)IS = ∇p(Eα⊗ Eα) =
∂p
∂zα ⊗ Eα
= Aα⊗ Eα ∈ Lin(R2, R3)
Nevertheless, since evidently in this case it does not lead to confusion the substitution
of Aα with Gα, we can recast the gradient of p as
P:= Gα⊗ Eα
which shows that indeed the tensor P belongs to the space Lin(R2, T
S). Without fear
of misunderstandings, where possible the vectors Aα, Aα and Gα, Gα will always
be confused.
Given any vector u = uαEα ∈ Ω, its image under the parametrization p is
repre-sented by the vector
t = Pu = P[uαEα] = uαP[Eα] = uαAα = tαAα ∈ TS
which incidentally shows that the cartesian components of the vector u coincide with the components of its image t under the parametrization p in the natural basis
{A1, A2}, that is
uα ≡ tα.
A vector field f defined on a surface S is called tangential if f ∈ TS at every
point of S; a tensor field L is called superficial if LN = 0 at every point of S and in particular tangential and normal if for any vector a and any point on S,
respectively La ∈ TS and La ∈ TS⊥.
The surface divergence of a surface vector field f : S → R3 is the field
divSf := tr(∇Sf) = (∇Sf) · IS : S −→ R.
Let fk = PSf and f⊥ = f · N with N ⊥ TS, then since
f = fk+ f⊥N
it results
∇Sf = ∇Sfk+ ∇S(f⊥N) = ∇Sfk+ N ⊗ ∇Sf⊥+ f⊥∇SN
hence, taking the trace we get
divSf = divSfk+ f⊥divSN.
The surface divergence theorem asserts that given a tangential field fk it results
Z S divSfkdA = Z ∂S fk· ν dL
with ν the unit normal to ∂S contained in the tangent plane TS; it further results
that Z S divSf dA = Z ∂S fk· ν dL + Z S f⊥divSN dA.
If L is superficial tensor field on S, then its surface divergence is the field defined by
a· divSL:= divS(LTa)
for every constant vector a in R3. Furthermore, since it results that LN = 0 we
deduce that Z S divSLdA = Z ∂S LνdL.
It is possible to show that all metric properties of a smooth surface S parameterized by p (such as measures of lengths, angles, areas with respect to those of the domain Ω) can be locally expressed in terms of the metric tensor of the surface, namely
A = PTP= (Aα· Aβ)(Eα⊗ Eβ) = Aαβ(Eα⊗ Eβ) ∈ PSym(Ω, Ω).
For example, given a vector t = Pu ∈ TS with u ∈ Ω a vector of modulus |u| = u,
its length can be calculated as follows:
|t| = [t · t]1/2 = [Pu · Pu]1/2 = [Au · u]1/2 = [Aαβuαuβ]1/2,
and, since tα = u
α, we obtain the expression of the so called First Fundamental
Form of the surface
|t| = [Aαβtαtβ]1/2.
In particular, the local area measure can be expressed as follows
dA = |A1× A2| = |PE1× PE2| = [det(PTP)]1/2 = [det A]1/2.
Let us now define a regular curve Γ on the surface S, introducing the parametrization:
λ ∈ [0, 1] ∈ R −→ z = c(λ) ∈ Ω −→ X = p(c(λ)) = ¯p(λ) ∈ Γ ⊂ S.
with
¯
p := p ◦ c.
The following set of definitions can now be given(see [?],pg...): ℓ(λ) = Z λ 0 | d¯p dλ| dλ = Z λ 0 |¯
p′| dλ (Curvilinear Abscissa or Arc Length)
T = d¯p dℓ/| d¯p dℓ| = d¯p dℓ = d¯p dλ dλ dℓ = ¯ p′
|¯p′| (Unitary Tangent Vector to Γ)
M = dT
dℓ /|
dT
dℓ | =
dT
dℓ /κM (Unitary Normal Vector to Γ)
where we denoted with ()′ = d()/dλ and used the fact that, by definition of arc
length, |d¯p/dℓ| = 1; we further introduced κM, the curvature of Γ in the plane
π(T , M ) spanned by the vectors T and M . The projection of the vector dT /dℓ on
respectively the definition of the so called normal curvature of the surface S and geodesic curvature of the curve Γ on the surface S:
Normal Curvature: κ = κ(T ) = dT
dℓ · N = κMM · N = κM cos θ(N , M )
Geodesic Curvature: kg = kg(Γ, N ) = |PS
dT
dℓ | = κM |PSM|
being PS = IS the perpendicular projector on TS. As a remark, notice that in general
M 6= N and thus the normal curvature generally differs from the curvature κM of
the curve Γ in the plane π(T , M ). Curves on the surface with identically equal to zero geodesic curvature are called geodesics.
Since it results that N ⊥ T for each point of S, thus in particular for each point of Γ, differentiating the relation T · N = 0 with respect of λ, we obtain that
T′ · N = −T · N′ where it results N′ = dN (¯p(λ)) dλ , T ′ = dT (¯p(λ)) dλ .
This allows to express the normal curvature as follows
κ = dT dℓ · N = dλ dℓT ′ · N = −dλdℓT · N′ = −T · N ′ |¯p′| = − ¯ p′· N′ |¯p′|2 .
Since the vector ¯p′ = |¯p′|T belongs to the tangent plane T
S, it results that ¯p′ ≡ PSp¯′, hence N′ = dN (¯p(λ)) dλ = (∇ ˜N)¯p ′ = (∇ ˜N)P Sp¯′ = (∇SN)¯p′ = −L¯p′
where - analogously to what has been previously done - we have let ˜N the 3D
ex-tension of N , and where the curvature tensor L := −(∇SN) has been introduced.
Substituting this relation in the expression obtained before for κ we obtain for the normal curvature of the surface in direction of the unitary tangent T :
κ := Lp¯
′· ¯p′
|¯p′|2 = L · (T ⊗ T ).
This last relation takes the name of Second Fundamental Form of the surface S. The curvature tensor is symmetric and tangential, that is
L = LT
LTN = 0 .
The proof of these properties can be elegantly given3 considering the surface S as
implicitly defined as the zero level of the function Φ, that is
S := {X ∈ R3|Φ(X ) = 0, Φ : R3 −→ R, such that a = |∇Φ| 6= 0}.
With this position the unitary normal vector to S reads as
N := ∇Φ
|∇Φ| =
∇Φ a
where the operator ∇ represents here the three-dimensional gradient. Taking the
gradient of the relation a2 = (∇Φ) · (∇Φ) we obtain
2a(∇a) = 2∇(∇Φ)T(∇Φ)
hence, considering that ∇(∇Φ) is indeed a symmetric tensor, we obtain that ∇a = (∇∇Φ)N .
Furthermore, taking the gradient of the relation aN = ∇Φ and using the last relation we get
a∇N = PS(∇∇Φ)
thus
L= −∇SN = −a−1PS(∇∇Φ)PS
which shows at the same time that L is tangential and it belongs to the space
Sym(TS, TS). Being L ∈ Sym(TS, TS), the Cayley-Hamilton theorem ensures the
existence of its real eigenvalues, the so called principal curvatures k1 and k2, and
the relative perpendicular eigenvectors, the principal directions of curvature of S. The Mean and Gaussian Curvatures are defined by
H = 1/2(k1+ k2) =
1
2trL
K = k1k2 = det L.
With these positions for the 2 invariants of the two-dimensional tensor L, the Cayley-Hamilton theorem yields
L2− 2HL + KIS = 0
which is a very useful relation to calculate the inverse of L, indeed
L− 2HIS+ KL−1 = 0 =⇒ KL−1 = 2HIS− L.
Note that the normal curvature of a surface in direction of the unit vector t is κ(t) = Lt · t.
After introducing the vector u ∈ Ω such that t = Pu, we can express the normal curvature as
κ = PTLP· (u ⊗ u) = L · (u ⊗ u) with L = PTLP
where the tensor L ∈ Sym(Ω, Ω) is actually the pull-back tensor of L on Ω. Since, as
we have shown previously, tα = u
α, this last relation can be expanded as follows
Taking the determinant of the relation L = PTLP we obtain that (see for example Do Carmo[13], pg.155, eq (4)) K = det L det A = L11L22− L212 A11A22− A2 12 .
It would be easy to check (see Do Carmo, [13], pg.232) that the cartesian
compo-nents Lαβ of the tensor L are expressible in terms of the cartesian components Aαβ
of the metric tensor A, which can roughly be summarized saying that
K = ˆK(Aαβ).
This fact leads to one one the major theorems in differential geometry, which will be presented in section 1.4.
1.3. Surface deformations, polar decomposition theorem. In what follows we introduce the concept of deformation of a surface. Let S and s be two surfaces parameterized as follows
S := {X ∈ R3|X = p(z ), z ∈ Ω ⊂ R2}
s := {x ∈ R3|x = g(z ), z ∈ Ω ⊂ R2}
with p and g two diffeomorphism on Ω. As done in the previous section, it is possible
to define on S and s the natural basis spanning the respective tangent planes TS
and Ts {A1, A2, A3} with Aα = ∂p ∂zα ∈ T S, A3 = N = A1× A2 |A1× A2| , {a1, a2, a3} with aα = ∂g ∂zα ∈ T s, a3 = n = a1× a2 |a1× a2| . The reciprocal basis on S and s can be defined as
{A1, A2, A3} with Aα ∈ TS, Ai· Aj = δij (hence A3 = A3 = N )
{a1, a2, a3} with aα ∈ Ts, ai· a
j = δj
i (hence a3 = a3 = n)
The surface gradients of the parameterizations p and g read, accordingly to the previous section, as follows
P= (∇p) = Aα⊗ Eα ∈ Lin(Ω, TS)
G = (∇g) = aα⊗ Eα ∈ Lin(Ω, Ts).
Let now A and B be two vector spaces with dim A = dim B,
then the following tensor spaces can be introduced (see Gurtin & Murdoch, [21])
Sym(A) := {S ∈ Lin(A)| ST = S},
PSym(A) := {S ∈ Sym(A)| S is positive definite},
Orth(A, B) := {Q ∈ Lin(A, B)| (QTQ) = IA, (QQT) = IB}
Rot(A, B) := {Q ∈ Orth(A, B)| det Q = +1}
InvLin(A, B) : {S ∈ Lin(A, B)|∃ S−1 ∈ Lin(B, A) such that S
−1S= I A
SS−1 = IB }.
The tensor S is called as the inverse of S, and the set InvLin represents the set of all invertible, linear operators between the two vector spaces A and B. When dim A 6= dim B then InvLin(A, B) is empty.
It is now possible to define by construction the inverse tensors of P and G
P−1 := Eα⊗ Aα where P−1P= (E α⊗ Eα) = IΩ PP−1 = (Aα⊗ Aα) = IS G−1 := Eα⊗ aα where G−1G= (E α⊗ Eα) = IΩ GG−1 = (aα⊗ aα) = Is
with IΩ, IS and Is the identity operators on the spaces Ω, TS and Ts respectively .
Let now f be a smooth function mapping points of S in points of s, such that
g = f ◦ p.
We can think of S as the reference configuration and s as the current configu-ration of S under the deformation f , that is
s = f (S).
By taking the gradient of the aforementioned relation g = f ◦ p, the chain rule yields
∇g = (∇f )(∇p) =⇒ G = (∇f )P =⇒ (∇f ) := GP−1.
Standing on this expression, we infer that (∇f ) so defined actually belongs to the
space Lin(TS, Ts), since as already noticed it results that G ∈ Lin(Ω, Ts) and P ∈
Lin(Ω, TS). Substituting the expressions obtained before for G and P−1, we deduce
the following explicit expression for (∇f )
(∇f ) = GP−1= (aα⊗ Eα)(Eβ ⊗ Aβ) = aα⊗ Aα ∈ Lin(TS, Ts),
whose inverse is rapidly checked to be
(∇f )−1 = Aα⊗ aα ∈ Lin(Ts, TS).
On the other side, it is evident that (∇f ) can be identified with the surface gradient of f ; indeed since the surface gradient is defined as
F:= (∇Sf) = (∇f )IS = (aα⊗ Aα)(Aβ ⊗ Gβ) = (aα⊗ Gα)
hence it is evident that F ≡ (∇f ) ∈ InvLin(TS, Ts). In some following parts of the
thesis, we will extend the domain of the surface gradient F by post-composition with the projector, that is
¯
F:= FP ∈ Lin(R3, Ts).
Very useful for the developments of this thesis is the following slightly generalized version of the polar decomposition theorem [21]:
Theorem 1 (Polar Decomposition Theorem). Each F ∈ InvLin(TS, Ts), admits
the unique decomposition
F= RU
being R ∈ Rot(TS, Ts), and U ∈ PSym(TS, TS). Moreover,
U2 = FTF= C, R = FU−1.
Analogously to the metric tensors PTPand GTG of the surfaces S and s, it would be
possible to show that the changes of all metric properties between S and s = f (S)
can be expressed in terms of the Green-Cauchy tensor C = FTF, which can be
explicitly calculated as
In particular, the local area variation of a neighborhood of a point X in S can be shown to be equal
J(X ) = da(f (X ))
dA(X ) = [det C]
1/2
where dA and da represent the area measures of infinitesimal neighborhoods of X ∈ S and of its image x = f (X ) ∈ s, and where the Jacobian J here represents the so called areal stretch.
1.4. Some fundamental facts from differential geometry. As first thing,
let us give some fundamental definitions. Let S ⊂ R3 be a surface. We say4 that a
point X ∈ R3 is a limit point of S if every neighborhood of X in R3 contains a
point of S distinct from X . S is said to be closed if it contains all its limit points.
S is bounded if it is contained in some ball of R3. If S is closed and bounded, it is
called compact. A surface S is said to be connected if any two of its points can be joined by a continuous curve in S.
Surface deformations that leave unaltered all the metric properties of a surface are called isometries.
Definition 1 (Isometries5 ). A surface deformation f : S → s is an isometry if
for all X ∈ S and all pairs of vectors (u1, u2) in TS(X) we have
u1· u2 = Fu1· Fu2.
It is evident that, within the context of classical continuum mechanics, we can rephrase the definition of isometry saying that a deformation f of a surface S is an isometry if
C= FTF≡ IS ∀ X ∈ S,
which substantially implies that F ≡ R = Rot(F) at every point of the surface. Such a deformation can be interpreted as pure bending of the surface S, where the length of all its fibers and the angles between any couple of its fibers remain locally unaltered. In particular, we trivially see that if f is an isometry then J = 1.
One of the major theorems in the differential geometry stands on the concept of isometries. Indeed, as already announced, the Gaussian curvature of a surface can always be expressed as a function of its metric properties. Indeed we have
Theorem 2 (”EGREGIUM” - Gauss). The Gaussain curvature K of a surface
is invariant by local isometries.
We shall now recall the Gauss-Bonnet theorem, which is ’probably the deepest the-orem in the differential geometry of surfaces’ [Do Carmo].
4See [13], pg.112
Theorem 3 (Gauss-Bonnet6). Let R ⊂ S be a region of an oriented surface S, with a closed regular boundary ∂R. Then
Z
RK dA = 2πχ(R) −
Z
∂R
kgdℓ,
K the Gaussian curvature, kg the geodesic curvature of the curve ∂R on the surface
S, and the number χ(R) the ”Euler-Poincar´e” characteristic of the region R.
The Euler-Poincar´e characteristic of a regular region R ⊂ S can be determined through the following rule
χ(R) = F − E + V
where F ,E and V respectively represent the number of triangles (F aces), sides (Edges) and V ertices of a triangulation of the surface R, which can be defined as follows
Definition 2 (Triangulation of a surface7). A triangulation of a surface R ⊂ S is a
finite family ℑ of triangles Ti, with i = 1, ..., n such that
(1) Sn
i=1Ti = R.
(2) If Ti∩Tj 6= Ø, then Ti∩Tj is either a common edge of Ti and Tj or a common
vertex of Ti and Tj.
It would be possible to show (see [13], pg.273) that the number χ(R) for a compact, connected surface is indeed related to the genus of the surface, that is the number of handles of the surface, through the relation
g = 2 − χ(R)
2 .
As a conclusive remark, let us recall the following theorem of global differential ge-ometry, which will be useful in this thesis.
Theorem 4. Let S be a regular, compact and connected surface with Gaussian
cur-vature K > 0 and mean curcur-vature H constant. Then S is a sphere.
1.5. How to reconcile with the classical notation. Let us now give a brief review of the classical index notation in order to compare it with the absolute notation adopted here.
Given the following parametrization for the surface S
p : z = (z1, z2) ∈ Ω ⊂ R2 =⇒ X (p) ∈ S ⊂ R3,
6See [13], pg.274, for a generalized version of this theorem to surfaces endowed of a non regular
boundary.
the following notation is classical for the derivative of p with respect to the cartesian
coordinate zα
p, α≡ ∂p
∂zα
hence the natural basis on a neighborhood of S reads as
Aα = p, α = PEα,
where the tensor P : Ω → TS has been previously introduced. Let now v be a smooth
vector field defined on S = p(Ω).
Letting ˆv = v ◦ p and omitting as usual point-wise dependence, the following
deriv-ative of ˆv w.r. to zα ˆ v, α:= ∂ ˆv ∂zα .
might be expanded, within the formalism adopted in this thesis, as follows ˆ
v, α = (∇v)p, α= (∇v)Aα= (∇v)ISAα = (∇Sv)Aα.
Aside the natural basis Aα it is convenient to introduce a reciprocal basis Aα,
spanning the same tangent plane TS, with the following property:
Aα· Aβ = δβα.
Any given vector v ∈ TS is expressible in both basis:
v = vαAα = vαAα
and its natural and reciprocal components are found according to
vα = v · Aα, vα = v · Aα.
The surface gradient of p has been explicitly calculated as
P= Aα⊗ Eα.
The components of the metric tensor A = PTP = (Aα· Aβ)(Eα⊗ Eβ) are usually
denoted by
Aαβ = Aα· Aβ
and the relative determinant J2 = det C is usually called
A = det(Aαβ).
Given a vector u = uαEα ∈ Ω, its image t under the parametrization p(z ) has norm
defined by the first fundamental form:
|t|2 = Aαβuαuβ.
The inverse of the metric tensor A can be computed as
A−1 = (PTP)−1 = P−1P−T = (Eα⊗Aα)(Aβ⊗Eβ) = (Aα·Aβ)(Eα⊗Eβ) = Aαβ(Eα⊗Eβ).
where the cartesian components Aαβ of the inverse of the metric tensor are called its
the reciprocal components, thus from
we have that
AαγAγβ = δβα.
This last relation relates the vectors of the natural and the reciprocal basis as follows
Aα= AαβAβ.
The directed normal N to the surface S is given by:
N = A1× A2
|A1× A2|
= A1× A2
J .
The components of the vectors Aα defined on the surface S
Aα, β = ∂Aα
∂zβ
are expressible in our formalism as
Aα, β = (∇SAα)Aβ;
the components of this vector in the tangent plane TS are usually denoted by the
Christoffel symbols
Γγαβ = Γγβα := Aα, β· Aγ = (∇SAα)Aβ· Aγ,
whereas the out-of-plane component is given by
Aα, β· N = (∇SAα)Aβ · N .
The curvature tensor of the surface S is defined as the surface gradient of the
directed unit normal to S,with reversed sign, that is L = −(∇SN) ∈ Sym(TS, TS).
The reciprocal components Lαβ of L are thus
Lαβ = L · (Aα⊗ Aβ) = −(∇SN)Aα· Aβ = −N, α· Aβ,
so that
L= Lαβ(Aα⊗ Aβ).
Differentiating the relation N · Aα= 0 with respect to zβ we obtain:
(N · Aα), β = N, β· Aα+ N · Aα, β = (∇SN)Aβ · Aα+ (∇SAα)Aβ· N = 0
thus the components of the curvature tensor can be written as
Lαβ = (∇SAα)Aβ · N = N · Aα, β
hence the derivative of the vector Aαwith respect to zβcan be written in the following
form
Aα, β = ΓγαβAγ+ LαβN.
The Mean and Gaussian curvatures can now be expressed as follows:
H = 1 2trL = 1 2L· (Aγ⊗ A γ) = 1 2A αβL αβ K = det L = 1 2[(trL) 2 − trL2] = 1 2[(A αβ)2(L αβ)2− LγδLµλAµδAλγ] = L11L22− L2 12 A
where it is easy to check that the last term can be rewritten as
K = e
αβeλµL αλLβµ
2A
being the cyclic index such that e12 = −e21= 1 and e11= e22= 0.
Let us consider a surface vector field v = v (x) = vαA
α. The derivative of v with
respect to zα is:
v, α = (vβAβ), α = v, αβ Aβ+vβAβ, α=vβ, α+ vγΓβαγ Aβ+vβLαβN = v; αβ Aβ+vβLαβN.
As it is immediate to check, letting PS = I − N ⊗ N = (Aγ⊗ Aγ) to be the local
perpendicular projector, it results
Pv, α = (Aγ⊗ Aγ)v, α= vγ
; αAγ
which means that the term vγ
;αAγ (usually referred to as the the covariant
deriv-ative of the vector field v ), is the projection of v, α on the local tangent plane. In
the literature sometimes the following notation for the components of the covariant derivative is adopted
vγ ; α = v
γ | α
depending on the choice of the domain where the field v is defined.
Let us finally introduce the concept of surface divergence of a superficial vector field v , that is simply the trace of the surface gradient of v
1.6. Notes of calculus of variations. We here recall some facts relative to the variational derivative of functionals which will be useful in the thesis.
a) Let Ω be an open domain in RN and let u(x) be a generic vector valued mapping of
Ω in RN. The most important necessary condition to be satisfied by any minimizer
of a variational integral F(u) =
Z
Ω
F (x, u(x), Du(x), D2u(x), ..., Dmu(x)) dΩ
with prescribed values of u on the boundary, is the vanishing of its first variation δF(u, ϕ) := lim ε→0 F(u + εϕ) − F(u) ε ε=0 ,
for all perturbations ϕ of u which do not change its boundary value. By means of the fundamental lemma of the calculus of variations, the vanishing of the first variation yields the differential Euler-Lagrange equations for u as necessary conditions to be
satisfied by minimizers of F which are of class Cm.
Cm-solutions u of the equation δF(u, ϕ) = 0 are called weak extremals. If such
solutions are even of class C2m, an integration by parts will lead to Euler’s equation,
the solution of which are called extremals.
The so-called null Lagrangians are some kind of degenerate variational integrands F for which the corresponding Euler equations are satisfied by any smooth function u with compact support. As an example, the integral
Z
Ω
div u dx
depends, via the Gauss-Green theorem, only on the assigned boundary values of u on ∂Ω, hence its first variation is vanishing for all perturbations with compact support. Another example of null Lagrangian already encountered in section (1.4) when treating the Gauss-Bonnet theorem, and it is represented by the total curvature of a surface S in the space
Z
S
K dA.
This integral will indeed play a crucial role for the aim of this thesis.
b) For future use within this thesis, let us consider the following class of variational integrals F(s, ˆu) := Z s ˆ F (x , ˆu(x ), ...) da
where ˆu : s → RN is a generic spatial vector field defined on points of the current
surface s of a reference surface S under a deformation f , ˆF is the variational
x ∈ s. Let us show that the first variation of the functional F can be written as follows δF = Z s [ ˆδF + ˆF divsy] da.
First of all, in order to give a precise definition of the first variation of the functional F(s, ˆu), which depends, as matter of fact, both on the field ˆu and on the geometry of the surface s, we map the functional on the reference configuration, hence
F(f , u) := Z
S
F (f (X ), u(X ), ...) J(X ) dA
where we have F = ˆF ◦ f and u = ˆu ◦ f to denote, respectively, the material
descriptions of the variational integrand ˆF and of the field ˆu; dA here represents the
area measure of a neighborhood of a point X ∈ S and J represents the Jacobian of the transformation f : S → s, defined as
J = [det C]1/2 = [det(FTF)]1/2
where
F= (∇Sf)
represents the surface gradient of deformation of f . The functional F now depends on two independent fields u and f , hence it is possible to define its first variation perturbing independently these fields, that is
δF := lim ε→0 F(f + εη, u + εϕ) − F(f , u) ε ε→0
where the perturbations η and ϕ are in general functions in C1
c(S, RN). Since the
reference surface S is actually independent of ε, we have F(f + εη, u + εϕ) =
Z
S
FεJεdA
where we have called Fε the perturbed variational integrand
Fε:= F (f + εη, u + εϕ, ...)
and Jε the perturbed Jacobian
Jε= [det Cε]1/2 = [det(FTεFε)]1/2.
The perturbed surface deformation gradient reads as
Fε = [∇S(f + εη)] = F + ε∇Sη.
hence
Jε= [det(F + ε∇Sη)T(F + ε∇Sη)]1/2.
For ε → 0, the perturbed Jacobian can be expanded in Taylor series around ε = 0
Jε = J + ε dJε dε ε=0 + o(ε),
where dJε dε = d dε(det Cε) 1/2 = 1 2(det Cε) −1/2d det Cε dCε · dCε dε = = 1 2(det Cε) −1/2(det C ε)C−Tε · F T (∇Sη) + (∇Sη)TF+ O(ε) = = 1 2JεC −T ε ·FT(∇Sη) + (∇Sη)TF+ O(ε) .
The well known fact that, given any invertible tensor A, it results that ∇A(det A) =
(det A)A−1 has been used (see Gurtin, [22], pg.23). Since given two generic tensors
A ∈ Sym and B it results that
A · B = A · 12(B + BT) =⇒ A · (B + BT) = 2A · B
we have that, since C−Tε is a symmetric tensor8
dJε
dε = JεC
−T
ε · [FT(∇Sη)] = Jε[Fε−1F−Tε ] · [FT(∇Sη)] + O(ε)
which, evaluated at ε = 0, using the definition of scalar product and the fact that tr(AB) = tr(BA), yields
dJε dε ε=0 = J[F−1F−T] · [FT(∇Sη)] = = J tr[F−1F−TFT(∇Sη)] = J tr[F−1(∇Sη)] = J tr[(∇Sη)F−1].
At this point we introduce the vector function y : s → R3, which represents the
spatial description of the perturbation η
y◦ f = η.
The vector y can actually be interpreted as a virtual displacement of points of the
current surface s, indeed letting xε to denote the position of the point x = f (X )
after the perturbation η, we have that
xε= f (X ) + εη(X ) = x + εy (x ) =⇒ y = [η ◦ f−1](X ) = lim
ε→0
xε− x
ε .
Taking the gradient of the relation y ◦ f = η (∇y)(∇f ) = (∇η)
and composing both members with the inclusion map on S, we get
(∇y)(∇f )IS = (∇η)IS =⇒ (∇y)F = ∇Sη,
where we remind that F ∈ Lin(TS, Ts), hence
FF−1 = Is
we finally obtain that
(∇y)FF−1 = (∇y)Is = (∇Sη)F −1 =⇒ (∇sy) = (∇Sη)F−1. 8By definition C−1 ε = [F T εFε]−1 = F−1ε F−Tε which is symmetric.
On the basis of this last relation, we can further expand the expression of the deriv-ative of the perturbed Jacobian w.r. to ε as follows
dJε
dε
ε=0
= J divsy,
where the definition of surface divergence has been used. For the sake of simplicity,
we will not stress the fact that the fields (∇sy) and divsy appearing in the last
relations are actually material descriptions of such fields, hence evaluated at a point
x = f (X ).
These last developments allow the Taylor expansion of Jε to be expressed as follows
Jε= J[1 + εdivsy] + o(ε).
Let us now go back to the expression of the perturbed functional Fε, which now reads
as
Fε =
Z
S
Fε[1 + ε divsy]J dA,
thus the first variation of F can be computed as follows δF = Z S lim ε→0 Fε[1 + εdivsy]J + o(ε) − F J ε ε=0 dA = = Z S lim ε→0 Fε− F ε + limε→0 o(ε) ε + Fεdivsy ε=0 J dA
In conclusion, we get the expression of the first variation of a functional defined on a non-fixed surface, expressed both on the reference surface S and on the current surface s δF = Z S [δF + F divsy]J dA = Z s [ ˆδF + ˆF divsy] da
where the first variation of the variational integrand δF and its spatial description ˆ
δF have been defined as
δF = lim
ε→0
Fε− F
ε , δF = (δF ) ◦ fˆ
Solution Theory
1. Classical solution theory of fluid mixtures.
The main constituents of lipid bilayers exhibit miscibility under suitable thermo-mechanical conditions. During the mixing process no chemical reaction occurs be-tween the different kinds of components, thus the mass of each one is always con-served. Deviations form ideal mixing (where components are poly-disperse, that is randomly distributed in the bulk), as well from total immiscibility (where compo-nents tend to segregate into disjoint domains within the bulk), yield the possibility of coexistence of different material phases - each one characterized by a different chemical composition - within the bilayer; this process is known as lateral phase separation in phospholipid mixtures.
After an introduction to the classical solution theory in absence of deformations, in this section we will explain the physical basis of the occurrence of phase separation phenomena in lipid bilayers. In Chapter 5 these concepts will provide the basis of our new chemo-mechanical model, where deformation effects are accounted.
1.1. Basic definitions. We give in this section a short list of basic definitions. Molecules. A molecule is the smallest particle in a chemical element or compound that has the chemical properties of that element or compound. Molecules are made up of atoms that are held together by chemical bonds. These bonds form as a result of the sharing or exchange of electrons among atoms.
Mole. The mole is the amount of substance of a system which contains a number of
elementary entities equal to the number of atoms in 0.012 kilogram of the isotope12C;
this number is known as the Avogadro’s number and it is equal to NA= 6, 022137·1023
elementary entities.
Atomic and Molecular Weight. Improperly referred to as a weight, it is the relative mass of an atom with respect of the unit atomic mass, equal to 1/12 of the
mass of the isotope 12C, that is u = 1, 66054 · 10−27kg; hence, atomic weight is an
a-dimensional number. The molecular weight is obviously the sum of the atomic weights of all the atoms which constitute the molecule.
Concentration. Generally speaking, concentration is the measure of how much a substance is mixed with other substances. There are many different ways to express quantitatively this kind of information, in particular we will deal with the following ones:
- Mole or Molar or Molecular Fraction. The mole fraction (or molar fraction) of a chemical species in a mixture is the ratio between the number of moles of the given species and the total number of moles of the mixture; being the number of moles proportional to the number of molecules through the Avogadro’s number, it is evident that molar fraction and molecular fractions coincide.
- Mass Fraction. Obviously, it is the ratio between the mass of a given substance and the mass of the whole mixture. The mass fraction does not coincide with the molar fraction of the species.
1.2. Basic thermodynamics - classical approach. In what follows we will briefly review the basic principles of thermodynamics in the simplified version adopted on elementary books of physics and chemistry (in particular we refer to the book by Silvestroni,[56]). Later on, all these concepts will be embedded in the more general and correct framework of Continuum Thermo-Chemo-Mechanics.
Thermodynamical System. It is defined as each perceptible quantity of matter which can be macroscopically observed, usually constituted by a (homo or heteroge-neous) set of material bodies. The words perceptible and macroscopic come along together: macroscopic variables, as pressure or temperature, can be defined (⇔ mea-sured) for systems which consist of a great amount of elementary particles and are, in this sense, macroscopically perceptible. We intend as material body a region of the space endowed with a continuous distribution of mass. We further define a ther-modynamical system to be isolated if there is no energy or mass transport across its boundary, closed if there is energy transport but not mass transport, and open if there are both energy and mass transport.
State Variables. Extensive and Intensive Quantities. We mean as state
vari-ables the smallest set1of independent variables which can describe the macrostate2
of a thermodynamical system. We will define as extensive all those state variables which depend upon some extensive properties of the system (for example its total mass or its number of moles, or its extension); extensive variables are always additive. On the other hand, intensive variables are all the remaining (for example pressure, temperature, molar fractions, etc...); intensive variables are not additive.
First and Second Principle. The simplified versions of both principles are the following:
i) First Principle: dE = d(Q + W )γ ∀ processes γ ;
ii) Second Principle: dQγ ≤ T dS ∀ processes γ ;
here E and S are respectively the internal energy and the entropy of the system, Q the heat entering the system during the process γ, W the mechanical work done on the system during the process γ and T the temperature. Both internal energy and entropy are state functions, in the sense that they are path-independent functions and thus can describe the state of a system independently of the path followed to reach it. With γ is denoted a generic - reversible or irreversible - process. For γ
1Here we embrace the concept of ”minimal state” treated in [12].
2While Classical Thermodynamics is interested in macrostates, Statistical Mechanics studies
the microstates, intended as the instantaneous configurations of all the elementary particles which constitute a system. Each macrostate corresponds to an enormous number of possible microstates, all experimentally undistinguishable among each other.
reversible, the equality holds in the second principle, hence the heat entering the system during a reversible process equals
dQRev = T dS.
For an open system, the expression of W has to account for both mechanical and for
mass diffusion work, W = WM + WD. The mechanical work done on a fluid system
by the external environment has the form dWM = −pdV , being p the pressure. From
now on, we will focus our attention on fluid systems.
State Functions. Beyond the state functions E and S, it is possible to introduce the following ones:
−Enthalpy: H = E + pV
[for closed systems, dH = dE + pdV + V dp = d(E − W ) + V dp =(p=cost)= dQp = cpdT ]
−Helmholtz Free Energy: Ψ = E − T S
[for open systems, dΨ = dE − T dS − SdT =(T =cost)= dE − T dS = dWRev]
−Gibbs Free Energy [or Free Enthalpy]: G = H − T S
[for open systems, dG = dE + pdV + V dp − T dS − SdT =(p=cost)(T =cost)= dW
D Rev]
These state functions are important in several problems of mechanics with thermal and chemical effects. In particular, as it will be shown later, the Gibbs free energy plays a crucial role during heterogeneous equilibria. The following mnemonic scheme is useful to remember the relations between the different state functions:
E → +pV → H
↓ ↓
−T S − T S
↓ ↓
1.3. Material and geometric phases. Let us assume that the phenomenon we are observing is completely described by a set C of independent fields defined on the domain occupied by the body. These fields describe purely geometrical evolutions of the body (that is, its motion) as well as all its non-geometrical evolutions (that is, all kinds of physical and chemical phenomenons which intrinsically depend upon some material characteristics of the body); granted the independence of the variables of the list C, it is possible to define unambiguously the both sets of geometrical and material variables as follows:
Geometrical Variables: The set CG ⊆ C of all purely geometrical variables.
Material Variables: The set CM = C/CG of all non-geometrical variables.
Evidently, CG∩ CM = {⊘} and CG ∪ CM = C. Since the two sets CG and CM are
disjoint, it is possible to focus our attention separately on each one; if we focus our attention on the evolution of material variables throughout the body, the following definition of Phase might be given:
A phase is a perceptible portion of matter whose material proper-ties are uniform or at most slowly and smoothly varying within its extension, and endowed at the macroscopical level with a boundary across which such properties have a discontinuity or a continuous and abrupt variation.
On the other hand, the viewpoint classically adopted in Continuum Mechanics is based on the recognition of geometric fields (such as,for example, the gradient of de-formation) as the variables whose discontinuities or abrupt variations define the oc-currence of a phase boundary, and thus the definition given above might be rephrased as follows:
A phase is a region of a body, characterized by uniform or at most slowly and smoothly varying gradient of deformation with respect to a given reference configuration, and which admits discontinuities or continuous and abrupt variations across the phase boundary.
Since the two viewpoints actually focus on disjoint properties of the material body, precisely material and geometrical, it appears legitim to us to label as material phases those recognized through the analysis of material properties, and as geo-metrical phases those recognized through the analysis of geogeo-metrical properties; moreover, the question whether or not the material phases are coupled with geo-metric phases, and how this coupling is achieved, is certainly non trivial and is the main topic of this thesis.
Phase Transitions between different phases can be roughly classified as first and second odred phase transitions. The first-order phase transitions are those that in-volve a latent heat. During such a transition, a system either absorbs or releases
a fixed (and typically large) amount of energy. Because energy cannot be instanta-neously transferred between the system and its environment, first-order transitions are associated with mixed-phase regimes in which some parts of the system have com-pleted the transition and others have not. Such mixed-phase systems are difficult to study, because their dynamics are violent and hard to control. Many important phase transitions fall in this category, including the solid/liquid/gas transitions and Bose-Einstein condensation.
The so called second-order phase transitions are the continuous phase transitions, mainly characterized by the absence of an associated latent heat. Continuous phase transitions are easier to study than first-order transitions, due to the absence of latent heat, and they have been discovered to have many interesting properties. The phe-nomena associated with continuous phase transitions are called critical phephe-nomena.
1.4. Heterogeneous equilibrium - The Gibbs phase rule. Let us consider
a material body3 B ⊂ E in which C different chemical species coexist in P different
material phases. Such a body is an heterogenous mixture. Within this simplified treatment of equilibrium of heterogenous mixtures, we will assume that no boundary layer exists within the different material phases, hence each material phase f =
1, ..., P is assumed to occupy a region Bf ⊆ B such that
B =
P
[
f =1
Bf and Bf ∩ Bk= {⊘} , ∀f 6= k.
Each Bf is an open thermodynamical system, since the mass of each specie can
rearrange in different phases during a process. If we assume that all constituents are
non reacting, which means that the mass of each constituent is conserved during
all processes, the following system of C constraints holds:
P
X
f =1
nfi = ni = const. ∀ i = 1, ..., C
where we denoted with nfi the number of moles of the constituent i in the phase f ,
and with ni the total number of moles of the component i in the mixture.
Chemical Potentials & the Gibbs Phase Rule. The chemical equilibrium of different chemical species coexisting in different phases within a region B of the space is analyzed via variational arguments, resembling the method adopted by Gibbs in his work on the equilibrium of heterogeneous substances (Gibbs,[17]).
Let us consider a fluid mixture B in which C different chemical species coexist, and let us make the assumption that the body is under thermal and mechanical equilibrium, that is:
−Mechanical Equilibrium : p = const. all over the body; −Thermal Equilibrium : T = const. all over the body.
We now want to investigate the conditions under which the species in the mixture re-arrange themselves in different material phases within the body B, during an isother-mal and isobaric process. The classical simplifying hypotesis of chemically homoge-neous phases will be assumed, that is each phase is characterized - beyond the same values of pressure and temperature - by uniform values of the molar fractions. Let us then make the following constitutive assumptions for the total internal energy and entropy of the body B:
E(B) = ˆE(p, T, n1, ..., nC);
S(B) = ˆS(p, T, n1, ..., nC).
3In this first simple exposition, we are not concerned about the distinction between the body
Both quantities are extensive and thus additive: this means that, since B is the union
of disjoint phase regions Bf, it results:
E(B) = E( P [ f =1 Bf) = P X f =1 E(Bf) = P X f =1 Ef(p, T, nf1, ..., nfC); S(B) = S( P [ f =1 Bf) = P X f =1 S(Bf) = P X f =1 Sf(p, T, nf1, ..., nfC).
where we introduced the list of extensive variables of each phase region, {p, T, nf1, ..., n
f C},
and we contextually admit that each material phase is characterized by its own con-stitutive equations. The number of moles of each chemical specie is given and thus the following C relations hold:
ni =
P
X
f =1
nfi ∀ i = 1, ..., C.
Let us denote - as it is done by Gibbs in his work - all perturbations of the state functions and variables with the symbol δ (see chapter 2 for further details on varia-tions). As proposed by Gibbs, the equilibrium configurations of a mixture B in contact with a heat bath at constant temperature T can be determined by imposing the vanishing of the total internal energy for all perturbations which leave unaltered the total entropy and the mass of each constituent in the mixture; it will be shown in section 3 of this chapter that this statement is equivalent to look for the solutions of the following constrained variational problem:
δ{Ψ(B) − L(B)} = 0
for all perturbations of the independent variables such that
ni(B) = ¯ni(B) = const. ∀ i = 1, ..., C.
The potential energy term Ψ represents the so called ballistic free energy (see also Ericksen [15], pg.7), namely in this case the Helmholtz free energy, defined as
Ψ := E − T S
with T the constant temperature of the surrounding heath bath. The work term L(B) can be expressed in the usual form −p dV valid for fluids; being furthermore p = const. we have: L(B) = Z B−p dV = P X f =1 Z Bf −p dV.
The constrained variational problem can now be recast in free form using Lagrange’s
multipliers; introducing the list of scalars
constant all over the domain, we will look for the solutions of the equation δF(B) = δ ( Ψ(B) − C X i=1 µi[ni(B) − ¯ni(B)] − L(B) ) = 0
for all possible perturbations of the variables {n1
1, ...nP1; ...n1C, ..., nPC}. Such
pertur-bations are now free, since the constraint on the conservation of the mass of each constituent has been embedded in the energy functional. Since p and T are known constants, and being without constraints
Ψ(B) = Ψ( P [ f =1 Bf) = P [ f =1 Ψ(Bf) = P [ f =1 Ψf, δni = P X f =1 δnfi, it results δF(B) = δΨ(B) + p P X f =1 δVf − C X i=1 µiδni = = P X f =1 [δΨ(Bf) + pδVf − C X i=1 µiδnfi] = P X f =1 [δ(Ψf + p Vf) − C X i=1 µiδnfi] = = P X f =1 [δGf − C X i=1 µiδnfi] = 0.
The state function G = Ψ + pV = E − T S + pV , already defined as the Gibbs Free
Energy, depends on the state functions E and S, on the intensive variables p and
T and on the derived extensive quantity V , hence it is an additive function and it
depends on the main set of independent variables {p, T, n1, ..., nC}. According to this
constitutive dependence, we can expand the term δGf as follows:
P X f =1 C X i=1 [∂G f ∂nfi − µi]δn f i = 0 ∀ variations δn f i.
Being the variations δnfi linearly independent, it is finally possible to determine the
Euler-Lagrange equations, which are represented by the following system of
con-ditions: µi = µ1i = µ2i = ... = µPi ∀ i = 1, ..., C. where we denoted by µfi := ∂G f ∂nfi
the Chemical Potential of the constituent i in the phase f . The equilibrium con-ditions obtained through the Euler-Lagrange equations can be stated as follows
In an heterogeneous mixture of different chemical species, the chem-ical potential of each species at the equilibrium is the same in all the phases in which the species appears.
Molar Densities. Let us now show that the molar densities of all the extensive
functions defined on a phase domain Bf, as well as the chemical potentials, admit a
general constitutive dependence on the following list of intensive variables {p, T, χf1, ...χ f C−1} where χfi = n f i nf = nfi PC i=1n f i
are the molar fractions of each constituent within the phase Bf. By virtue of the
hypothesis of chemical homogeneity of each phase of the mixture and by thermo-mechanical equilibrium, we infer that, for any extensive function
M = ˆM (p, T, nf1, .., nfC)
defined on the phase domain Bf and for any scalar α > 0 it results that the function
M is 1 − homogenoeus, that is to say
α ˆM (p, T, nf1, .., nfC) = ˆM (p, T, αnf1, .., αnfC).
On taking in particular α = 1/nf with nf the total number of moles in the phase
Bf, we get that the molar density of the function M depends solely on the molar
fractions, being m := M nf = ˆM p, T, nf1 nf, ..., nfC nf ! = ˆM (p, T, χf1, ..., χfC). Nevertheless, because of the constraint
nf = C X i=1 nfi =⇒ 1 = C X i=1 χfi
we infer that, say, χfC can be expressed as function of the other C − 1 molar fractions,
hence
m = ˆm(p, T, χf1, ..., χfC−1).
This result shows that molar densities of each phase energy depend solely on the list
of independent variables {p, T, χf1, ...χ
f
C−1}, such as for example the molar density of
the Gibbs free energy of the phase f
gf := G f nf = ˆg f(p, T, χf 1, ..., χ f C−1).
Considering the definition of chemical potential of the constituent i in the phase f
and being nfi = χfinf we also get
µfi = ∂G f ∂nfi = ∂(nfgf) ∂χfi ∂χfi ∂nfi = n f∂gf ∂χfi 1 nf = ∂gf ∂χfi = ˆµ f i(p, T, χ f 1, ..., χ f C−1).
The chemical potential of the pure constituent i in the phase f for given values of the pressure and temperature
µ0fi := ˆµfi(p, T, δ1k, ..., δik, ..., δ(C−1)k).
are named standard chemical potentials and express the free energy of formation of a single mole at the given values of pressure and temperature of the considered constituent.
Variance of a system. We can now determine the number of independent intensive parameters in a mixture of C heterogeneous constituents in thermal, mechanical and chemical equilibrium in P different phases. The total number of intensive parameters is given by the constant values of pressure and temperature and the C − 1 molar fractions for each phase, thus a total number of 2 + P (C − 1). On the other side, at equilibrium we have the following relations between the chemical potentials of each constituent in each phase
µ1i = µ2i = ... = µPi ∀ i = 1, ..., C
which means a total number of C(P − 1) independent relations. This means the number of independent intensive parameters at equilibrium equals
V = 2 + C − P
and such number is usually referred to as the variance of the system. The definition of the variance of a thermodynamical system defines the Gibbs’ Law or the Phases’ Rule. For example, in a system in which water coexists in both liquid and vapor phases, the number of independent variables is V = 2 + 1 − 2 = 1.
1.5. Gibbs-Duhem equations - The common tangent rule. We have al-ready shed evidence on the fact that homogeneity of the Gibbs free energy implies that
Gf(p, T, α nf1, ..., α nfC) = α Gf(p, T, n
f
1, ..., nfC) ∀ α > 0.
Differentiating with respect to α and setting α = 1 we obtain the following relation:
Gf = C X i=1 µfinfi being µfi = ∂G f ∂nfi = ∂gf ∂χfi which, differentiated, yields:
dGf =
C
X
i=1
Calculating the total differential of the relation Gf(p, T, nf1, ..., nfC) = Ef + pVf − T Sf we obtain ∂Gf ∂p dp + ∂Gf ∂T dT + C X i=1 µfidnfi = dE + p dVf − T dSf + Vfdp − SfdT = = Vfdp − SfdT + dWRevD,f
where the term dWD,f
rev represents the infinitesimal reversible diffusion work.
Inde-pendence of the variations dp and dT yields that
Vf = ∂Gf/∂p, Sf = −∂Gf/∂T
thus we obtain the following expression for the reversible diffusion work in the phase f dWRevD,f = C X i=1 µfidnfi.
This allows us to write down the variation of the Gibbs free energy in the form
dGf = Vfdp − SfdT +
C
X
i=1
µfidnfi which, combined with the relation obtained before
dGf =
C
X
i=1
(dµfinfi + µfidnfi).
yields the following Gibbs-Duhem relations, valid for each phase
Vfdp − SfdT =
C
X
i=1
nfidµfi ∀ f = 1, ..., P
which establish a series of P constraints among the differentials of the intensive
variables {p, T, µf1, ..., µ
f
C} of the mixture. Dividing for nf the expression of Gf in
terms of chemical potentials and mole numbers we obtain that
Gf nf = C X i=1 µfi n f i nf ⇒ g f = C X i=1 µfiχfi.
Finally, also the Gibbs-Duhem equations can be rewritten using molar fractions,
simply dividing for nf the relation obtained before
vfdp − sfdT =
C
X
i=1
χfi dµfi ∀ f = 1, ..., P.
![Figure 5. Images experimentally obtained by Baumgart, Hess & Webb [7],](https://thumb-eu.123doks.com/thumbv2/123dokorg/7266141.82753/7.918.184.717.135.263/figure-images-experimentally-obtained-baumgart-hess-amp-webb.webp)
