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4. Rheology, Lubrication and Relaxations
Index
4. 1 Basic Rheological Concepts......................................................................................... 2
4.1.1 Stress-Strain and Strain-Rate Relationships........................................................... 2
4.1.2 Equation of Motions............................................................................................... 4
4.1.3 Viscous Flow in the Solid-Liquid Boundary Regime ............................................ 7
4.2 Lubrication – A Nanorheological Endeavor ................................................................. 9
4.2.1 Hydrodynamic Lubrication .................................................................................... 9
4.2.2 Extended Regimes of Lubrication........................................................................ 12
4.2.3 Viscoelastic Theory and Lubrication ................................................................... 14
4.3 Polymer Rheology....................................................................................................... 16
4.3.1 Linear Viscoelasticity........................................................................................... 16
4.3.2 Mechanical Models Summary.............................................................................. 18
4.4 Glass Transition........................................................................................................... 20
4.4.1 The Nature of the Glass Transition ...................................................................... 20
4.4.2 Molecular Mobility in Supercooled Liquids ........................................................ 24
4.4.3 Molecular Mobility from Liquid to the Glassy Solid........................................... 25
4.4.4 Applicability of the Molecular Mobility Models ................................................. 25
4.4.5 Molecular Mobility in Glassy Solids ................................................................... 26
4.4.6 Molecular Diffusion in Amorphous Polymers ..................................................... 27
4.4.7 Methods Used to Determine Glass Transition Temperatures .............................. 28
References and Additional Literature ....................................................................... 31
Rheology deals with the deformation and flow of condensed matter under the
influence of externally acting forces. Condensed matter of interest are fluids, complex
liquids such as colloidal systems, polymer melts, solids and glasses that show rate
dependent deformation properties. Rheology plays a major role in the understanding
of fundamental transport properties, e.g., in
the relaxation behavior of polymer materials,
the hydrodynamic properties of fluids,
the reaction dynamics involving fluids and interfaces,
the lateral dissipative forces that take place during a relative sliding
process of two bodies in contact, and
the deformation properties of ductile material.
Thus, transport properties and material properties are linked by virtue of the
rheological material properties. In Nanotechnology, where material constraints
imposed by interfaces strongly affect the transport properties, nanorheological studies
are imperative. This Chapter is addressing basic rheological concepts in terms of
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equations of motions, stress-strain rate relationships and relaxation properties. Further
bulk system rheological responses are compared those of confined systems.
4. 1 Basic Rheological Concepts
Searching in popular dictionaries, such as such as Webster's Encyclopedic
Unabridged Dictionary, for definitions of "solid" and "liquid", one finds rather
peculiar descriptions. For instance, in Webster's Dictionary one finds for solids the
definition "having the interior completely filled up", and for liquids the description
"composed of molecules, which move freely among themselves but do not tend to
separate like those of gases". Although amusing, Webster's Dictionary actually
expresses very well our common perception of matter.
Material or process descriptions are used for classification purposes with the
intent to simplify the degree of complexity. Inaccuracies, however, lead to
misconceptions. That shows particularly for systems that differ significantly from
thermally equilibrated bulk systems, such as nanoconfined material arrangements.
Instead of describing the material with a material phase property that demands
the system to be thermally equilibrated, it is better to describe the material's resistance
and relaxation properties in the presence of external stresses (forces per unit area).
Here, of particular importance is the rate dependence. A material that exhibits
perfectly solid-like behavior can be described with a simple stress-strain relationship,
and a material with perfectly liquid-like behavior shows a simple stress- strain-rate
dependence.
Solid-like materials deform and relax instantaneously with changes in the
applied external stresses. Hooke's Law is a manifestation of a solid-like behavior.
Thus, a perfectly solid-like behavior is synonymous with perfectly elastic.
Mechanical energy is stored in a perfectly elastic material without exhibiting any
form of energy dissipation. The energy is instantaneously regained with the discharge
of the external stresses. Note with this definition of a material behaving perfectly
solid-like, no structural arrangements, such as for instance "crystallinity", were
imposed.
The counter behavior to perfectly solid-like is perfectly liquid-like as found in
a Newtonian liquid, which will be introduced in the next paragraph. In general, any
realistic liquid and solid matter will behave in a mixed manner, solid-like and liquidlike, depending on the degree and time scale over which external stresses are acting.
To describe realistic material we will discuss in the next paragraphs stress- strain-rate
relationships in fluids. Our discussion will start with the interplay between stress and
strain rate in Newtonian and non-Newtonian fluids.
4.1.1 Stress-Strain and Strain-Rate Relationships
The basic equation of simple flow is described one-dimensionally by
Newton’s law of viscosity,
dv
(1)
τ yx = − ηo x ,
dy
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which relates proportionally the shear force per unit area, τxy, to the negative of the
local velocity gradient with a constant viscosity value, ηo.1 Liquids and gases that
follow this law are called Newtonian fluids. They are incapable of storing mechanical
energy. Experiments have shown that homogeneous nonpolymeric liquids, such as for
instance water, methane, toluene, and glycerol are well described by the Newtonian
model. Many industrially important liquid materials, however, do not or only partially
follow Newton’s law of viscosity, and are hence called non-Newtonian liquids. The
subject of non-Newtonian flow is a subdivision of the science of rheology. A more
generalized form of equation (1) is provided by
dv
(2)
τ yx = − η x
dy
where η is a function of either the shear or the velocity gradient.2 Many empirical
models have been developed over the past to describe experimental data. The fluids
were described in new terms, such as pseudoplastic, dilatant, and Bingham-plastic as
illustrated in Figure 9. Pseudoplastic (shear thinning) fluids include polymer solutions
or melts, greases, starch suspensions, biological fluids, detergent slurries, paints, and
dispersion media in various pharmaceuticals. In regions of pseudoplastic behavior the
apparent viscosity η decreases with increasing rate of shear. An opposite behavior is
observed for dilatant fluids, such as potassium silicate in water or solutions
containing high concentrations of powder in water.
Figure 1: Shear diagram for Newtonian and non-Newtonian fluids:
(a) Bingham-plastic; (b) pseudoplastic (also called shear thinning);
(c) Newtonian; and (d) dilatant.
The Bingham-plastic behavior is maybe the most intriguing of the nonNewtonian manifestation, because it combines, in a very simple manner, elastic
1
2
x denotes the direction of momentum and y the direction of momentum transfer.
vx denotes the velocity in x-direction.
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behavior with viscous Newtonian behavior. Bingham plastics exhibit solid-like (rate
independent) stress-strain behavior for stresses below some critical stress τo, and
liquid-like Newtonian stress- strain-rate behavior above τo. The model has been found
useful for many fine suspensions and pastes, such as soap, paper pulp and chocolate.
So far only time-independent fluid behavior has been introduced and
differentiated. There are additional types of non-Newtonian behavior under unsteadystate conditions: thixotropic fluids, rheopectic fluids, and viscoelastic fluids.
Thixotropic and rheopectic fluids exhibit a reversible decrease and increase,
respectively, in shear stress with time at a constant rate of shear. Examples of
thixotropic fluids are some polymer solutions and paints, and rheopectic fluids certain
sols. Viscoelastic fluids exhibit elastic recovery from flow deformations, which will
be discussed in more detail below because of their importance for polymers.
4.1.2 Equation of Motions
Any fluid flow where the mass is conserved can be described by two
equations, the equation of continuity (or mass conservation) and the equation of
motion. The equation of continuity, which describes the rate of change of density for
an observer who floats along with the fluid, is
Dρ
= −ρ( ∇ ⋅ v) = −ρdivv ,
(3)
Dt
in which the operator D/Dt is the substantial time derivative, ρ is the density of the
fluid at a fixed point, and v is the velocity vector.3,4 The equation of motions is
Dv
ρ
= −∇p − [∇ ⋅ τ] + ρg , (Cauchy Equation)
(4)
Dt
which is a statement of Newton’s second law in the form of mass times acceleration
per unit volume, on the right of equation (4), and the sum of three forces (the pressure
force ∇p , the viscous force, ∇ ⋅ τ , and the gravitational force, ρg )5 per unit volume,
on the left. The Cauchy equation is a Governing Equation such as the continuity
equation. The challenge in using Cauchy's equation is to find an appropriate
constitutive equation that relates the stress tensor, σ = pΙ + τ (combining both
compression and shear tensor), 6 with the strain Γ and/or the strain rate tensor
dΓ/dt ≡ Γ .
For an isotropic perfectly solid-like deformation the stress-strain relationship can
be expressed as (see Chapter on Continuum Mechanics)
 ∂f 
1



(5)
σ ij = 
= Kε kk δ ij + 2G ε ij − δ ijε kk  .
 ∂ε 
3


 ij  T =const.
where σij and εij represent the components of the stress and strain tensors,
respectively, K the modulus of compression and G the modulus of rigidity (or shear
3
Vectors are denoted in this text in either bold phase or with the vector symbol (e.g., v, v ).
4
The gradient and divergence are expressed with the del-operator; i.e.,
5
Double underlining is used for second-order tensor notation (e.g., shear tensor τ with elements τij).
Ι represents the identity tensor.
6
∇p and ∇ ⋅ τ , respectively.
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modulus). The strain and strain rate components are related to the deformation ui and
rate of deformation dui/dt ≡ uD i as
ε ij =
1  ∂u i ∂u j 
+
; i = 1,2,3 .
2  ∂x j ∂x i 
(6)
εD ij =
1  ∂uD i ∂uD j 
+
; i = 1,2,3 .
2  ∂x j ∂x i 
(7)
The strain rate tensor can be expressed by the dyadic product of the gradient and the
deformation velocity ∇v .7 It is:
[
]
t
d 1 ΓD ≡ ∇v + (∇v ) , with v = u .
(8)
2
dt
Analogous to the stress-strain deformation above, the dissipative stress tensor
components, σ*ik,, assuming as purely stress-strain rate relationship, can be expressed
as
∂Φ
1


σ*ik =
= ηiklm ε lm = ςε ll δ ik + 2η ε ik − δ ik ε ll  ,
(9)
∂ε ik
3


where η and ζ are viscosity coefficients. η represents the shear viscosity coefficient,
and ζ is the molecular resistance to isotropic expansion. Usually η is dominating ζ,
and thus, ζ is neglected. Φ replaces the free energy, f, and is called the dissipation
function. Neglecting ζ and assuming incompressibility, the constitutive equation (9)
can be written in the form
t
τ = 2ηΓ ≡ η ∇v + (∇v ) .
(10)
[
]
Substituting equation (10) into the Cauchy Equation and further assuming constant
viscosity, the equation of motion simplifies to the famous Navier-Stokes equation
Dv
ρ
= −∇p − η∇ 2 v + ρg ,
(11)
Dt
which describes the fluid flow of a Newtonian liquid.
For many liquids, however, the assumption of a constant viscosity (i.e., rate
independent viscosity) is not appropriate. Any realistic fluid is generally nonNewtonian. The strain rate dependence of the viscosity is expressed as
1
1
2
η = η(| ΓD |), with | ΓD |=  (ΓD : ΓD )
(12)
2

where | ΓD | represents the magnitude of the strain rate tensor not unlike the definition
for the length of a vector: i.e., α:β = ΣiΣjαijβji. Hence the constitutive equation for a
realistic fluid is
7
The Dyadic Product of two vectors is a special form of a second-order tensor;
e.g.:
 ∂
 vx
 ∂x
∂
∇v ≡  v x
 ∂y
 ∂
 vx
 ∂z
∂
vy
∂x
∂
vy
∂y
∂
vy
∂z
∂

vz 
∂x 
∂
vz 
∂y 

∂
vz 
∂z 
The transpose of a tensor is designated by
(∇v ) .
t
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1
1
2
τ = 2η(| Γ |)Γ , | Γ |=  (Γ : Γ ) ,
(13)
2

which lead to the generalized momentum equation of Navier Stokes applicable to
both, Newtonian and non-Newtonian fluids
Dv
ρ
= −∇p − η(| Γ |)∇ 2 v + ρg .
(14)
Dt
The magnitude of the strain rate tensor is provided in Table (1).
Table 1:
C|
Γ ≡ |Γ
Typical examples for the strain rate dependent viscosity, η(| ΓD |) , are provided
below for three models; the Bingham Model, the extensively used Power Law Model,
and the Carreau Model (used for polymeric liquids such as solutions and melts).

for τ≤τo
∞

τo
Bingham Model:
,
η= 
η +
 o 2 | ΓD | for τ > τ o
where τo defines a critical stress, above which the shear
behavior is Newtonian, described by a constant
viscosity ηo.
Power Law Model: η = m( 2 | ΓD |) n −1 n, m are fitting constants.
The material behaves shear thinning (pseudoplastic) or
shear thickening for n < 1 or n > 1, respectively.
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[
]
n −1
2
η − η∞
Carreau Model:
= 1 + (2λ | ΓD |) 2 ηo, η∞, λ, n are fitting constants.
ηo − η∞
For pseudoplastic materials n is typically between 0.2 and
0.6. λ is a characteristic relaxation time constant for the fluid
(i.e., the product 2λΓ represents the ratio of the
characteristic relaxation time and the characteristic time of
flow). ηo and η∞ represent two Newtonian limits for 2λΓ«1
and 2λΓ»1, respectively.
4.1.3 Viscous Flow in the Solid-Liquid Boundary Regime
Unsteady flow and viscous flow in boundary layers are of particular interest in
tribology. Start-up and control problems are demanding the study of transient
phenomena that occur when a fluid near a wall is suddenly set in motion. Under
heavy load the liquid boundary layers are responsible for the viscous drag of
lubricated systems.
Unsteady viscous flow near a solid flat surface, Figure 2, can be expressed
with the following differential equation:
∂2 v
∂v
ρ x = η 2x ,
(15)
∂y
∂t
that was derived from eqs. (3) and (11), assuming a constant density and viscosity,
and a sudden surface motion in x-direction with a velocity vsurf.
Figure 2:
Unsteady viscous flow of a fluid near a solid flat surface at sudden motion.
Equation (15) represents the classical one-dimensional diffusion equation. With the
boundary conditions set to
vx = 0
at t < 0, vx = 0 for all y,
vx = vsurf at y = 0 for t > 0, and
at y = ∞.
vx = 0
an the solution,
ξ
−
1
2
 η 
2
2
vx
= 1−
e−ξ dξ ; ξ ≡ y 4 t .
∫
v surf
 ρ 
π0
(16)
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The integral is the well known error integral, abbreviated as erf(ξ) and provided in
Table 2. Its asymptotic behavior, erf(∞) = 1, allows one to define a characteristic
thickness, called boundary layer thickness δ∗ as the distance y for which vx has
dropped to 0.01×vsurf. Hence, a boundary layer thickness of
η *
δ* = 4
t ,
(17)
ρ
can be found. The time t* represents a characteristic time of the momentum diffusion
through the boundary layer. Low viscous and high viscous fluids will significantly
affect the variable ξ, and hence the error signal and the boundary layer thickness.
Table 2: Error Function (plotted on the left)
It is reasonable to assume that the boundary layer is insignificantly growing
for t > t* and finite body sizes. If two bodies, separated by a fluid film, are in relative
motion, two sliding regimes can be differentiated, (a) a turbulent sliding regime; the
thickness of the confined fluid film is larger than δ∗, and (b) a laminar sliding regime;
the thickness of the confined fluid film is smaller than δ∗.
The relative sliding motion allows one to assume that the lower body is at rest
while the opposite upper body is sliding at constant velocity vsurf. Any frictional
losses in the turbulent sliding regime of a Newtonian liquid are due to velocity
fluctuations which give rise to turbulent shear stresses. The friction force related to
turbulent flow past a flat surface can be approximated as:
−
1
 Lv ρ  2
v2
turb
Ffric
= 0.072ρ turb WL turb  ,
(18)
 η 
2
with the plate area WL and length L and the flow velocity vturb of the turbulent fluid
lubricant [1]. Many semi-empirical theories have been developed over the years using
various expressions for the turbulent momentum flux, also called Reynolds stresses,
which will not be further discussed.
Within body distances smaller than the boundary layer thickness, momentum
is transferred from the upper body to the lower body. Blasius’ numerical solutions of
flow near the leading edge of a flat plate provide a drag force per surface of
la min ar
Ffric
= 0.664 ρηLW 2 v ∞ 3
(19)
where v∞ is the flow velocity far away from the solid boundary [2].
The laminar regime of lubricated sliding is, in literature, referred to
hydrodynamic lubrication and is extensively studied with the Reynolds Theory which
is discussed in the following paragraph.
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4.2 Lubrication – A Nanorheological Endeavor
4.2.1 Hydrodynamic Lubrication
As long as a laminar flowing lubricant film can support the load between two
sliding surfaces, the fluid can be considered to be in the hydrodynamic regime. The
opposing surfaces must be conformal for hydrodynamic lubrication. The study of
hydrodynamic lubrication is the study of a particular form of the Navier-Stokes
equations, or more generally the equation of motion - the Reynolds equations. The
Reynolds equations contain, as parameters, the viscosity, the density and the film
thickness. These three parameters can vary locally and depend on temperature,
pressure fields and the elastic behavior of the bearing surfaces. The following
assumptions are made to reduce the equation of motion for Newtonian fluids to the
Reynolds equation:
(a) The height of the fluid film y is very small compared to the dimensions of the
contact area,
(b) the pressure is constant across the fluid film,
(c) the flow is laminar, i.e., no turbulence occur,
(d) the inertia of the fluid is small compared to the viscous shear (examples of
inertia forces are fluid gravity and acceleration of the fluid),
(e) the fluid velocity at the bearing surfaces is zero (no-slip condition), and
(f) no external forces act on the film.
In general form, i.e., valid for compressible and incompressible Newtonian
fluids, the Reynolds equation is
∂(ρh )
∂  ρh 3 ∂p  ∂  ρh 3 ∂p 
∂
+ 6ρh ( U1 + U 2 ) + 12ρV
(20)

+ 
 = 6( U1 − U 2 )
∂x  η ∂x  ∂z  η ∂z 
∂x
∂x
with the film thickness, h, the sliding direction and its perpendicular surface direction,
x and z, respectively, the velocity of the two bearing surfaces in x-direction, Ui, the
radial component of the velocity, V, and the pressure, p, which is the mean of the
diagonal shear elements, τii. Only in the case of a very particular geometry, such as a
journal bearing, there is a general analytical solution of the equation (20).
Between two moving bearings, the lubricant flows in x-direction because of
pressure flow and shear flow, and in z-direction because of pressure flow only.
Hence, shear stresses can be introduced for a Newtonian fluid as
1 ∂p
µ
τx = −
(21a)
2 y − h ) + ( U 2 − U1 ) ,
(
2 ∂x
h
1 ∂p
τz = −
(21b)
(2 y − h ) ,
2 ∂z
The total drag, defined as
F = ∫∫ τdA ,
exerted by the moving bearing surface is, at y=0 or y=h,
z x
µ
 1 ∂p

F = ∫ ∫ ±
h + ( U 2 − U1 ) dxdz ,
h
2 ∂x

0 0 
(22)
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since Fz is perpendicular to the moving direction.
The challenge is now to derive from the Reynolds equation an analytical
expression for the pressure distribution. Assuming steady loading, incompressible
lubricants and a simple plane slider (which is a model for a one-dimensional thrust
bearing), Fig. 3, the Reynolds equation reads
dp − 6 ηU( h − h o )
=
,
(23)
dx
h3
where ho is the height of maximum pressure. As illustrated in Figure 3, the film
thickness, h, can be expressed as
h ( a − 1)
(24)
h = αx = 2
x.
L
Figure 3: Plane slider - a one-dimensional model for thrust bearings.
Equations (21), (22) and (23) lead to expressions for the shear stress and the
drag force
4h − 3h o
τ = ηU
,
(25a)
h2
2
2
Nh 2 2(a − 1) ln a − 3( a − 1)
L
F=
= ηUW Cf ,
(25b)
L 3( a + 1) ln a − 6(a − 1)
h2
with the width of the bearing, W, the normal load exerted by the fluid, N,
2
2
h1
 L
6ηUWL2  1  
2( a − 1) 
N = W ∫ pdx =
= ηUW   C p ,
(26)

 ln a −
h2
h 2 2  a − 1 
a + 1 
 h2 
the inlet and outlet height ratio, a,
h
(27)
a= 1 ,
h2
and the parameters of maximum pressure
2 h1h 2
2a
(28a)
ho =
=
h ,
h1 + h 2 1 + a 2
3ηUα( a − 1)
h − h2
po =
.
(28b)
; α= 1
2αa
L
The dimensional coefficients Cp and Cf are measures for the load capacity and
friction, respectively, of plane sliders. A maximum load capacity can be found by
2
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setting dW/da=0 which yields an inlet and outlet height ratio of a=2.2, and hence a
maximum load of
2
 L
N max = 01602
.
(29)
( ηUW ) h  C p
 2
and a resulting drag force of
h N
Fmax = 4.7 2 .
(30)
L
If the load capacity of a hydrodynamic bearing is exceeded, the lubricant wedge
separating the bearing surfaces will eventually diminish in volume.
A friction coefficient, µ, can be introduced as the ratio between drag force and
normal load
F h C
µ= = 2 f .
(31)
N L Cp
In Figure 4, the load capacity and the friction coefficient are plotted as a function of
the height ratio. The friction coefficient reaches its minimum at a = 2.55.
Figure 4: Load capacity and friction coefficient.
In the case the plane slider in Figure 3 slides parallel with distance h with very
slow relative velocity U, the pressure distribution can be neglected and the drag force
directly related to the momentum transfer, i.e.:
η
F = ULW ,
(32)
h
The load, N, is constant provided by the weight of the slider. This is, of course, a very
simplified solution of a parallel slider where entry effects, i.e., flow near the leading
edge of a flat plate as described in equation (11), are entirely neglected. We will later
see how this equation was used for liquid gap distances orders of magnitude smaller
than the contact area when we discuss surface forces apparatus measurements.
Calculations with other shaped sliders showed the height ratio to be very
important but find the effect of the shape of the lubricant insignificant for load
capacity and drag force determinations. A film shape of
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(33)
h = eβx ,
which provides exact solutions to the Reynolds equation for constant viscosity is,
therefore, justified. Because of the lengthy expressions of the exact solutions that can
be found in any theoretical hydrodynamic book [3] only the one-dimensional solution
is presented. Following the procedure above the load capacity and drag force is
3ηUWL2  a 2 − 1 a 2 ( a − 1) ln a 
N=
−
,
(34)
a 3 − 1 
( ah 2 ln a) 2  6
and
F=
4
ηUWL 3  ( a − 1) 

,
h 2 2a ln a  a 2 − 1 
respectively, with a load capacity maximum value at a=2.3 of
2
 L
N max = 0165
. ηUW   .
 h2 
(35)
(36)
4.2.2 Extended Regimes of Lubrication
As shown above, it is very challenging to attain solutions to the Reynolds
equation even if the configuration is very simple and any lubricating artifacts are
neglected. Features of lubricating artifacts are, for instance, striation with incomplete
lubricant, surface roughness, or elastic deformations of the bearings. Most of the
calculations above assume constant viscosity, which is however known to change
with temperature and under high pressure. If the contacting surfaces are
counterformal (i.e, non-conforming as assumed for hydrodynamic lubrication), local
pressures in the contact zone will be very high - up to several Gigapascal. Examples
of non-conforming contact regimes are shown in Figure 5.
Figure 5: Devices of non-conforming contact regimes.
Very high local pressures in the film cause the viscosity of the lubricant to
increase with the tendency to expand the film thickness over the predictions of the
hydrodynamic theory. Also, the elastic deformation of the bearing surface has to be
considered. High pressure lubrication is approached with an extended hydrodynamic
theory - the theory of elastohydrodynamic lubrication (EHL). Pressure spikes and
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sharp constriction of the film in the exit region of sliding bearings are poorly
investigated.
The viscosity of oil, as one of the most common lubricants, shows a fairly
close exponential relationship,
η = ηo exp( αP) ,
(37)
with the hydrostatic pressure, P, and the constant parameters, ηο (viscosity at zero
pressure), and α.8 The pressure coefficient α is of the order of 10-8Pa-1 for typical
mineral oils. While in a hydrodynamically lubricated bearing any increase in the
viscosity will be only a few percent, in the EHL regime the viscosity can increase by
over 20,000 times, and that at atmospheric pressure. Under these circumstances the
liquid can show a solid-like behavior.
Roughness, sliding velocity and pressure determine when full fluid lubrication
begins to break down and lubrication enters new regimes. The mixed regime of
lubrication is reached when in addition lubricant films, adhering to the surface
contours, should be considered. If there is no bulk liquid left and the lubricant is
reduced to a ultrathin layer, a few molecular layers thick, boundary lubrication comes
into effect.9 In reality, high pressure lubrication is due to roughness of the bearing
surfaces found in a mixed regime, assuming there is no severe wear. Mere boundary
lubrication is rather academic of nature, however, can be the dominant factor in
mixed lubrication, especially if the number of contact asperities is high. The
likelihood of asperity contact is expressed as the ratio between the minimum film
thickness of the bulk fluid and the r.m.s. roughness. The different regimes of
lubrication are very nicely illustrated with the Stribeck curve, which is also known as
the Reynolds-Sommerfeld curve, Fig. 6.
Figure 6: Stribeck Curve (schematic: (a) dry contact regime, (b) boundary lubrication, (c)
mixed lubrication, (d) elasto-hdydrodynamic lubrication, (e) hydrodynamic lubrication. The
friction force F is the product of the normal load and the friction coefficient.
The Stribeck curve relates the friction force (F=µN) to the hydrodynamic drag forces
of equation (32), i.e.,
8
9
Empirical relationship: α≈(0.6+0.96510lgηο)×10-8, [ηο]=[cPoise]
Note that ‘boundary lubrication’ should not be confused with ‘viscous boundary layer’.
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η bS
U,
(38)
h
where S is the apparent surface area, and ηb is the liquid bulk viscosity. The regime in
which hydrodynamic drag forces and friction forces correspond to each other is
known as Couette flow regime. It is recognized from various experiments with
surface forces apparatus (see below) that the bulk viscosity ηb in equation (38) has to
be replaced for rough surfaces by the effective viscosity ηeff∝(U/h)-2/3 (4-7). Note that
this famous power law has been observed for "simple liquids", i.e., liquids which are
independent on the shear rate (8-10). There is however a controversy of how
universal this power law is. In a theoretical work of Urbakh, Klafter and co-workers
it has been concluded that the exponent can vary between -2/3 and -1.0 (9, 11) in the
shear thinning regime which has also been experimentally observed by Israelachvili
et al. (12). An exponent of -1 leads to a velocity independent friction force in
equation (38) which may be contrasted with the behavior of a purely Newtonian
liquid where the viscous friction force is proportional to the velocity. It is however
important to note that the liquids studied by Israelachvili and co-workers were
complex fluids, i.e., fluids in which the shear forces are velocity or frequency
dependent as it will be discussed below.
F=
4.2.3 Viscoelastic Theory and Lubrication
As discussed above, fluids can show non-Newtonian behavior in which the
viscosity is either a function of the shear stress or the shear rate. One special kind of a
non-Newtonian fluid behavior is the viscoelastic behavior. A viscoelastic fluid
exhibits both viscous flow and elastic restoring forces. All real liquids show
viscoelastic behavior if stressed fast enough. The elastic response to stress is a fluid
property such as the viscosity. As per our discussion in Chapter 3 (Continuum
Mechanics) or equation (5) the stress in a solid body can be expressed as
τ = Gγ ,
(39)
where τ and γ replace σik and 2εik, respectively, to express simple shear
deformation.10 Hence, the stress rate behavior of a solid is
dτ
dγ
=G .
(40)
dt
dt
Considering the case of a slow stress disturbance, i.e. a stress rate that is much lower
than the inverse of the relaxation time t* of the fluid, the fluid shows predominantly
viscous behavior (i.e., Newtonian, no strain is built up). On the other hand, the fluid
behaves more like an elastic body if the stress rate is fast. Based on this asymptotic
stress behavior, the following extended differential equation,
1 dτ 1
dγ
+ τ=
,
(41)
G dt η
dt
provides a reasonable description for viscoelastic fluids, where G is the shear
modulus and η the viscosity. The first and second term on the left side of the last
expression describes the rate of elastic and flow deformation, respectively.
10
uii=0 and σii=0 for simple shear.
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Considering the rate of deformation dγ/dt=0, after a fluid element has been rapidly
deformed and constrained in its deformed shape, it yields that the stress exponentially
decays with time. This is expressed by
 G 
τ = τ o exp − t .
(42)
 η 
The ratio t*=η/G is called the Maxwell relaxation time. Assuming again a bearing
configuration as discussed above, the differential equation for viscous fluids can be
rewritten as
d 2 p  G  dp 12G  Uh − q 
+
=
(43)



dx2  ηU  dx
U  h3 
with
h 3  1 dp U d 2 p 
q = Uh − 
+
.
12  η dx G dx 2 
This expression corresponds to the Reynolds equation (20) if 1/G is set equal to zero
and the Reynolds equation is reduced to one-dimension.
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4.3 Polymer Rheology
4.3.1 Linear Viscoelasticity
In the previous paragraphs, the basic ideas of elasticity theory of solids and
viscous behavior of fluids have been discussed. Viscoelastic properties of fluids have
been introduced in the special case of thin liquid films under high pressure. In
addition to fluids, solids can also behave viscoelastically under mechanical stresses.
Polymeric materials, in particular, can show viscoelastic behavior because of their
molecular chain structure. Based on the Boltzmann’ s superposition principle, we
consider a linear superposition of the present and past deformations by describing the
stress as follows:
t
dγ
τ( t ) = ∫ G( t − t ') dt' .
(44)
dt'
−∞
This equation is the corresponding integral equation to equation (40) with the
relaxation function G(t-t’) as the replacement for the shear modulus. The inverse
representation is
t
dτ
γ ( t ) = ∫ J ( t − t ') dt' ,
(45)
dt
'
−∞
where J(t-t’) describes the deformation as response to the stress. Fully viscous or
elastic behavior is achieved by setting τ(-∞)=0 or γ(-∞)=0, respectively. If viscous
flow is neglected then the time-dependent functions retain only elastic relaxation
components. Therefore, as t→∞, the stress and the deformation become proportional
to one another, i.e.,
τ
(46)
G∞ = ∞ .
γ∞
The mechanical stress-strain behavior can then be described, with the help of
relaxation times t*, as
dγ 1
= ( γ − γ ) , and
(47a)
dt t *γ ∞
dτ 1
= ( τ − τ)
dt t *τ ∞
(47b)
or, by substituting equation (46)
dτ
dγ 

= G∞  γ + t *γ  .
(48)

dt
dt 
In the case of a one-time-disturbance in form a step function, equations (44)
and (45) can be replaced by
τ( t ) = γ oG( t ) , and
(49a)
γ ( t) = τ oJ( t) .
(49b)
J(t) and G(t) resemble a creep function or a relaxation function, respectively, Fig. 7.
τ + t *τ
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Figure 7:(a) Relaxation, (b) Creep.
In the case of a periodic disturbance, the strain can be rewritten in the form
γ=γosin(ωt) which substituted in equation (44) leads to
τ( t ) = γ o [G \ sin( ωt ) + G \ \ sin( ωt ) ] ,
(50)
with
∞
∞
G = ω ∫ G(s )sin(ωs )ds , and G = ω ∫ G (s) cos(ωs)ds .
\
\\
0
(51)
0
Thus, the response of the system is constructed of an in-phase and out-of-phase
component. The two harmonic functions can be replace by
τ 
τ 
G \ =  o  cos δ , and G \ \ =  o  sin δ ,
(52a)
 γo
 γo
with
G\\
J\\
(52b)
= tan δ = \ ,
G\
J
which introduces the phase relation (loss tangents δ) between the disturbance and the
response. The energy, W, that is dissipated during a viscoelastic deformation is
T
T
dγ
W = ∫ τdγ = ∫ τ dt = πγ oG \ \ = πγ oG \ tan δ .
(53)
dt
0
0
An oscillating load can also be mathematically expressed in a complex notation, so
that the following identities and equations apply for
• complex stress
τ( t ) = τ oe iωt ,
(54a)
γ ( t ) = γ oei( ωt − δ ) ,
τ*
η* = * ,
γ
•
complex strain
•
complex viscosity
•
complex relaxation function G* ( t ) =
τ*
1
,
* = *
J ( t)
γ
(54b)
(54c)
(54d)
with the relation
(54e)
G* = G \ + iG \ \ , and, J * = J \ + iJ \ \ ,
\ \\ \
\\
to the in- and out-of-phase components G ,G ,J and J . Many relationships follow
from these equations. For instance,
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1 / G\
J =
,
1 + tan 2 δ
1 / G\\
\\
J =
−1 ,
1 + ( tan 2 δ )
\
(55d)
(55d)
G\ \
G\
.
(55d)
,
η\ \ =
ω
ω
Finally, by substituting equations (49a) and (49b) into equation (48) with Gt→∞≡Gω→0,
the shear stress and modulus is
t *γ
τ( ω → ∞ )
*
*
τ(1 + iωt τ ) = G o γ 1 + iωt γ , and, G ∞ =
= Go * .
(56)
γ ( ω → ∞)
tτ
Since G∞>Go, the characteristic time for the creep process is longer than the
relaxation time.
η\ =
(
)
4.3.2 Mechanical Models Summary
Based on the discussed linear viscoelastic theory of stress-strain and rate
behaviors, the following summary for the possible behaviors of materials can be
obtained:
(a) fully elastic behavior:
τ = Gγ ,
dτ
dγ
= η , or
(b) fully viscous behavior:
dt
dt
d
τ
dγ 

= G∞  γ + t *γ  ,
(c) linear viscoelastic behavior: τ + t *τ

dt
dt 
Simple phenomenological models have been developed by using mechanical springs
and dampers to describe Hooke’s elasticity and Newton’s viscosity, respectively.
Maxwell’s and Kelvin-Voigt’s models. in their simplest form. are sketched in
Figure 8.
Figure 8: Viscoelastic models: (a) Maxwell model, (b) Voigt Model.
The models contain a single spring with a damping term either attached in series or in
parallel. The significant difference between the two basic mechanical viscoelastic
models is, that the system based on the Kelvin-Voigt model relaxes to a finite
displacement in the event of a step-like disturbance, and the Maxwell model not. For
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t→∞, the Kelvin-Voigt model behaves fully elastic, and the Maxwell model fully
viscous. Hence, the two models are only of limited use in describing the behavior of
real systems. The Maxwell model cannot account for the time-dependent aspect of
creep, and the Kelvin-Voigt model fails to explain stress relaxation.
The Kelvin-Voigt model corresponds to the following choices of variables:
• the strain γ is the observable variable,
• the stress is the strain’s associated variable, and
• the stress is divided into an ‘elastic’ part τe=Gγ and an ‘inelastic’ one τi=ηdγ/dt
so that the total stress is τ=Gγ+ηdγ/dt.
The Maxwell model corresponds to the following choices of variables:
• the strain γ is always the observable variable,
• the stress is the strain’s associated variable, and
• the strain is divided into an ‘elastic’ part dγe/dt=dτ/dt*1/G and an ‘inelastic’ one
dγi/dt=τ/η so that the total stress is dγ/dt=dτ/dt*1/G +τ/η.
Setting the stress equal to a periodical function (equation (49a)) and using equation
(56) and its counterpart for ω→∞, equations of modulus and compliance can be
derived which is summarized in Table 3.
Table 3: Two Basic Viscoelastic Models
Kelvin-Voigt
Maxwell
dγ
dγ
Diff. Equation: ηo
+ Goγ = τ
Diff. Equation: τ = G o
dt
dt
−t / t*
G( t ) = G
G( t ) = Ge
(
J( t ) = J 1 − e− t / t
G \ ( ω ) = G,
*
)
2
(
ωt * )
G ( ω) = G
2
1 + (ωt * )
G \ \ ( ω ) = ωη
tan δ = ωt * , t * =
J* =
J( t) = J + t / η
\
η
G
G \ \ (ω) = G
1
(
G*
(relationship between complex compliance
and complex modulus)
J\ =
J
1 + (ωt * )
J\\ = J
2 ,
ωt *
1 + (ωt * )
2
ωt *
1 + (ωt * )
1
η
tan δ = * , t * =
ωt
G
1
1
J \ (ω) = J = , G \ \ (ω) =
G
ωη
2
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4.4 Glass Transition
To describe material behavior or transport properties it is essential to know, besides
structural properties, also kinetic properties such as the molecular mobility. Particularly
in amorphous solid systems, such as polymers, the determination of the molecular
mobility is of foremost importance. A parameter that has been found to be very useful in
predicting the degree of molecular mobility is the glass transition temperature.
The glass transition is defined as the reversible change in an amorphous material
(e.g., polystyrene) or in amorphous regions of a partially crystalline material (e.g.,
polyethylene), from (or to) a viscous or rubbery condition to (or from) a hard and
relatively brittle one [4]. The "midpoint" temperature at which (or the temperature
regime over which) the transition occurs is defined as the glass transition temperature.
The term glass transition is used in the materials community pervasively, implying that it
describes a well-understood material phenomena or material property. However, similar
to other poorly defined terms, such as friction, a large ambiguity consists. The problem
in describing the glass transition or in reporting the transition temperature unambiguously
arises from:
• "diverging" instrumental methods used to determine transition values
• critical parameters (such as, for instance, measurement rates or areas)
• the many processes that can be responsible for changing molecular mobilities,
and which can lead to diverging theoretical models and interpretations.
4.4.1 The Nature of the Glass Transition
The definition of the glass transition presented above seems to depend on our
perception of terms, such as "solid", and "liquid". One can group materials into "solids"
and "liquids" by either considering the materials rheological response, or by analyzing
the thermodynamic phase of the system.
A rheological material description is concerned about stress – strain and stress –
strain-rate relationships. As a solid-like behavior we describe a rheological process that is
expressed with a purely stress – strain relationship. Controversially, a liquid-like behavior
is a purely rate-dependent process and cannot be described with a stress – strain
relationship, but with a stress – strain-rate relationship. Any real material will exhibit, to
different degrees, both behaviors depending on (a) its microscopic (molecular) and
macroscopic intrinsic mobility, and (b) the extrinsic stress rate to which it is exposed.
Considering free energy changes between equilibrium states one can identify the
solid- and liquid-phase by a discontinuity in the first partial derivatives of the Gibbs free
energy, G, with respect to the relevant state variable (e.g., temperature, T, and pressure,
P), as illustrated for the volume-temperature plot in Figure 9. Discontinuities, as
expressed in the first partial derivatives of the Gibbs free energy
 ∂G 

 =V
 ∂P  T
 ∂G 

 = −S
 ∂T  P
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 ∂ (G / T ) 

 = H ,
 ∂ (1 / T )  P
are found in the volume-temperature relationship, the entropy S and enthalpy H.
 ∂G 

 =V
 ∂P  T
T
Tm
Figure 9: Volume discontinuity. First-order transition between liquid and solid.
(Tm melting temperature).
First mesophases to solid and liquid phases that are found in polymeric systems are
the states of a "glass" and a "melt". The glass state is known to exist also for many nonpolymeric materials. From a structural viewpoint a solid can be either crystalline,
amorphous (unstructured) or partially amorphous-crystalline. A "glass" is an amorphous
solid and can exhibit both solid- or liquid-like behaviors. The melt behaves rheologically
liquid-like, exhibiting like the amorphous solid a short-range order not existing in the
liquid phase. More intermediate condensed phases have been reported in materials to
exist as illustrated in Figure 10. The different melt-states compared to their corresponding
glass-state show the same structure but exhibit different "large amplitude" molecular
motions, such as translational, rotational and conformational motions. "Large" amplitude
motions ("large" relative to vibrational thermal motions) operate on the picosecond
(10-12s) timescale. Around the glass transition temperature the timescale of this large
amplitude motion is slowed to milliseconds or even seconds.
Empirically it has been found that for many glasses with mobile units (called beads)
of the size of one to six atoms the heat capacity increases "abruptly" by about
11 J/(K mol) at the glass transition temperature. Discontinuity in the heat capacity are
known to exist and can be caused by second-order transitions. Second order transitions
are known as order-disorder transitions (found also at the onset of ferromagnetism), and
express a continuous behavior of the free energy and its partial derivatives, and a
discontinuous behavior for the second partial derivative with respect to the relevant state
variable. Hence there are no discontinuities in S, V or H at the transition temperature but
there are discontinuities in:
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- Heat Capacity, Cp
 ∂ 2G 
 ∂S  C p
−  2  =   =
 ∂T  P  ∂T  T
∂  ∂ ( G / T )  
 ∂H 
  = 
 = CP

∂T  ∂ (1 / T )  P  P  ∂T  P
- Compressibility, κ
 ∂ 2G 
 ∂V 
 2  = 
 = −κV
 ∂P  T  ∂P  T
- Therm. Expansion Coeff., α
 ∂  ∂G  
 ∂V 
 ∂T  ∂P   =  ∂T  = αV
T  P 
P
 
First and second order transitions are illustrated in Figure 11. If compared to property
changes in glasses around the glass transition temperature, one finds some similarity
between the glass transition and the second order transition. There are however
significant differences. Cp, κ and α values are always smaller and closely constant below
the glass transition temperature, Tg, compared to the values above Tg. This is clearly in
contrast to the second-order transition. A more disturbing finding is that Tg
measurements are highly heating/cooling rate dependent, Fig. 12, which does not occur
for a "true" second-order transition.
Fig. 10: On the right hand side: Transitions between the various crystal and melt phases.
On the left hand side: Degree of disorder, in terms of entropy change ∆S, found in
freezing conformational, orientational, and positional motions.
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Figure 11: Schematic representation of the changes with termperature of the free
energy and its first and second derivatives for (a) first order, (b) second order and (c)
glass transition.
Figure 12: Depending on the cooling rate any liquid can freeze into a glass phase (fast
quenching; e.g., metallic glasses). In polymers, the transition from a melt to a glass is
not discontinuous. Hence the assignment of a single transition value for Tg is
ambiguous.
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4.4.2 Molecular Mobility in Supercooled Liquids
Between the solid glass state and the liquid melt state, i.e., for temperatures between
Tg and the melting temperature, Tm, the material can be treated as a supercooled fluid,
Figure 12. Using Eyring's thermodynamic model that applies to supercooled liquids we
can find an expression for the viscosity in the form of:
kT
 ∆G a 
η = η0 exp
; η0 =
C Eν a
 kT 
with the height of free enthalpy barrier to be crossed by Eyring's jump, the Boltzmann
constant k, the activation volume of the stress deformation (σ) va (assume vaσ << kT),
and a constant CE. With decreasing temperature (i.e., T → Tg) Eyring's Arrhenius
behavior brakes down, and the viscosity is represented by a power law, or by the
empirical law by Vogel-Fulcher-Tammann:
 A 
.
η = η0 exp
 T − To 
The relationship between modulus, G, and viscosity, i.e., η=Gτ, introduces a
characteristic time τ, which is expressed with the free volume theory as
 ∆α −1 

τ = τ o exp 
*
T−T 
where ∆α is the difference in the volumetric expansion coefficient (typical value 5 × 10-4
K-1), T* is the critical temperature below which the free-volume is zero (i.e., no diffusion
is possible). The Einstein-Stokes equation relates the critical free-volume temperature to
the glass transition temperature as T * ≈T g −50K .
The free-volume theory (by Cohen and Turnbull) is restricted to weak interactions of
neighboring structural units. The theory is extended to strong interactions with statistical
fluctuation theories (by Gibbs and Di Marzio and by Adam and Gibbs). Again a
characteristic time can be assigned to the transitions probability between different
configurations; i.e.,
 ∆µ ln 2 
1

τ = τ o exp 

 ∆C p  T ln  T 
T 
 2
where ∆µ is the barrier of free enthalpy per structural unit, opposing the co-operative
rearrangement, ∆Cp is the difference between specific heat of the liquid and that of the
glass, and T2 is the temperature for which the configurational entropy becomes zero if the
supercooled liquid could be maintained at thermodynamic equilibrium up to this
temperature.
Comparing the two theories one finds that the free volume theory presumes that the
entire motion of atoms results only from the distribution of free volume without crossing
the energy barrier. Contrarily in the fluctuation theory, one assumes that the atomic
motions are the consequence of a co-operative rearrangement of an assembly of structural
units under the effect of thermal fluctuation allowing the jump of the energy barrier
separating the initial configuration from the final configuration. It has been argued very
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difficult (or impossible) to establish with experiments a preference for either of the two
theories.
4.4.3 Molecular Mobility from Liquid to the Glassy Solid
Using the theory of entropy fluctuations the characteristic time of molecular
movement can be described by the following Arrhenius law:
∆µk ln 2
 E 
τ = τ o exp   ; E =
∆C p ln (Tf / T2 )
 kT 
with an apparent activation energy E, and the imaginary (fictive) temperature Tf
characterizing the structural state of the glass. The existence of a critical temperature T2
is still in dispute. A different approach by Adam and Gibbs, which is based on the
concept of defects and molecular mobility, avoids the issue of critical temperatures with
the expression
τ
E 
τ = o exp  av  ; E av = ∆µ C + kT ln (C d ) ≈ ∆µ C .
d
d
Cd
 kT 
Cd (defect concentration) is playing the role of an order parameter that is linearly related
to the configurational entropy. Thermodynamically one can express the entropy as
T
S(T ) = N A ∆C p ln   ,
 T2 
where NA is the Avogadro's Number, ∆Cp is the difference between specific heat of the
liquid and that of the glass, and T2 is the temperature for which the configurational
entropy becomes zero if the supercooled liquid could be maintained at thermodynamic
equilibrium up to this temperature.
4.4.4 Applicability of the Molecular Mobility Models
With Vogel-Fulcher-Tammann description of a supercooled fluid we assumed that the
configurational state of the liquid changes such that thermodynamic equilibrium is
maintained. This is only possible if the molecular mobility during the experiment is
sufficient for attaining this equilibrium. If this is not the case, the configurational state of
the liquid remains constant, and the liquid is frozen, i.e., a glass.
As discussed above three distinctive temperature regimes can be distinguished:
• T > 1.2 Tg: The molecular mobility is sufficient to maintain the distribution of
free volume and entropy is in thermal equilibrium (supercooled fluid model).
• Tg < T < 1,2 Tg: The viscosity is high, on the order of 105 Pa⋅s. Fluctuation
models and defect models are more appropriate.
• T < Tg. Defects are frozen and their concentration Cd constant. Thermal
equilibrium does not apply anymore. The system is described as non-ergodic.
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4.4.5 Molecular Mobility in Glassy Solids
In experiments of amorphous solids it has been found that there is not only one value
of a characteristic time τ. Although it was found that the properties of glassy solids vary
over time, leading to a distorted power law, which is expressed with the Kohlrausch
relaxation function
F(t) ≡ [X(t)-X(∞)]/[X(0)-X(∞)] = exp[-(t/τ)b], 0<b<1.
X is the property that is relaxed. The extended exponential b is a structural parameter. It
increases with increasing disorder in the matter. Generally one discusses the various
characteristic times in terms of parallelly and independently occurring microscopic
processes, and assigns to them degrees of freedom with a given statistical weight.
A different approach by Palmer suggests that the processes occur in series,
representing a well-determined microscopic origin that correlates the various degrees of
freedom. A given molecular motion is dependent on the availability of other degrees of
freedom of mobile neighboring structural units. The model by Palmer goes back to the
Ising's spin model, in which changes in discrete spin levels depend on the condition of
the neighboring spin level. It could be shown that Palmer's model can simulate a function
of relaxation involving an extended exponential as shown above with the Kohlrausch
factor. The characteristic times τ(n) are gradually increasing with the spin level, n, and
are limited by a finite maximum value τmax= τ(n=∞) (ergodic limit).
It is important to note that in an experiment of thermal activation, the critical time of
the experiment texp decides the system response. If texp > τmax, the system behaves in an
ergodic manner, and thermodynamic laws apply. Contrarily, if texp < τmax the thermal
evolution cannot be described by classical statistical thermodynamics generating
metastable configurations (see Figure 13 below).
Figure 13: Schematic variation of free enthalpy G with temperature for the three
states of matter: gas, crystal and liquid. Equilibrium between phases at TF, Ts, TE:
melting, sublimation and boiling temperature, respectively. The glassy state is
outside equilibrium.
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4.4.6 Molecular Diffusion in Amorphous Polymers
The transport of matter in condensed phases is related to the molecular mobility. The
coefficient of diffusion is related to the characteristic time τ of molecular mobility by
λ2
,
τ
where λ is the dimension of the structural unit (i.e., distance between neighboring units).
For T > Tg and T < Tg, the following two relationships are appropriate
D=
 B ∆µ 
D = D o exp  − s 
 C d kT 
−
and
1
 τβ  b
D = D o  − 1− b  ,
 to 
respectively, where Bs is a constant expressing the minimum configurational entropy for
transition, to corresponds to the molecular vibration, and τβ is the characteristic time of
the "crankshaft" type of rotation.
Considering small-molecule diffusion, the coefficient of diffusion can be expressed in
terms of the Einstein molecular friction coefficient, ξ, as:
kT
,
D=
ξ
which varies with temperature by a WLF law (William-Landel-Ferry) as:
C1WLF (T − Tg )
ξ = WLF
; C1WLF ≈ 17.44 ; C 2WLF ≈ 51.6 K .
C 2 + (T − Tg )
The empirical WLF law can be deduced from the diffusion equation above for T>Tg ,
which yields approximate values for ∆µ of 0.2 eV, Cd(Tg) of 0.1, ∆Cp of 28 J mole-1 K-1.
If we consider now autodiffusion, i.e., the diffusion of within the same species of
molecules, we can distinguish between the following diffusive mechanism in a
macromolecular system:
• Diffusion of repeat units: D1=kT/ξ1 (ξ1 is the monomeric coefficient of
friction),
• Diffusion of short chains: D=kT/Nξ (N is the number of repeat units)
• Diffusion of long chains: D=D1/N2 (for N>Nc; Nc being the critical number of
repeat units in the chain at which Tg becomes independent of molecular
weight).
The molecular diffusion in dilute and glassy polymers depends essentially on the
forces of intermolecular interactions. We can introduce a characteristic time for reptation,
τrep, which is related to the monomeric characteristic time as:
2
Rg
3
τ rep = τ monomer N =
; R g = N 1 / 2 λt ,
D trans
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where Rg is the radius of gyration of the macromolecular coil, and Dtrans is the
translational diffusion through a tube of length L=Nλt. Notice that the viscosity η ∝Gτrep
is affected by the power 3 of the molecular weight (experimentally: η ∝N3.4).
4.4.7 Methods Used to Determine Glass Transition Temperatures
The most classical method for obtaining Tg are calorimetric measurements
(differential scanning calorimetry, DSC) that record the specific heat capacity as a
function of the temperature, Cp(T). This methods provides various distinct temperatures,
which could be used to define Tg, as schematically illustrated in Figure 14.
Figure 14: Schematic thermogram Cp(T). Shows various distinct temperatures.
Either of them could be used to define Tg. The schematic identifies the most
commonly agreed value for Tg
Other methods involve dilatometric measurements for the determination of the
specific volume, Figure 15, mechanical property measurements (thermomechanical
analysis, TMA (Figure 15), and dynamic mechanical analysis, DMA (Figure 16)), and
dielectric measurements.
In DMA measurements, as illustrated in Figure 16, it is not possible to identify halfdevitrification as in calorimetry and dilatometry (see Fig. 14 and 15), so one usually
chooses the peak in the loss tangent, tanδ=G''/G', to determine Tg. Varying the frequency
of DMA as well as the temperature, it is possible to establish activation energies and
relaxation time spectra. This is illustrated in Figure 17 for poly(vinyl acetate), PVAC.
The Tg peak values of the tanδ curves are shifted to a higher temperature with frequency.
A shift of about 5-7 oC was found to correspond to a one decade increase in frequency.
Similar results were obtained on polystyrene and polycarbonate. The apparent activation
energy, ∆Ηα, defines the amount of energy required to develop molecular mobility in the
polymer chains (includes carbon-carbon bonds, segmental chain bond motion, and inter
molecular separations between polymer chains). A graph of frequency versus reciprocal
glass transition values in kelvin, Fig. 18, combined with the following relationship2
∆H a
log f = log f o −
2.3RT
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is used to determine the activation energy of long chain motions as provided in Table 4.
Currently, these data are however difficult to link to the underlying cooperative
molecular motion. It is important to note that additional maxima in tanδ (not shown here)
are possible and caused by coupling of local molecular motion with chosen type of
deformation.
Figure 15: Dilatometry (top) and TMA (bottom) of an epoxy printed circuit
board.
Figure 16: DMA of unplasticized poly(vinyl chloride) (CH2-CHCl-)x. Glass transition (β-process or
main dispersion) at 354 K. G'' exhibits also high dispersion transition at lower temperature, referred to
as γ processes. The jumping of particular rotational isomers, the so-called "kinks", give rise to
γ processes.
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Figure 17: Varying the frequency of DMA as well as the temperature for PVAC
Fig. 18: Activation diagram (based on Figure 17 for PVAC).
Table 4: Apparent activation energies
Polymer
Apparent Activation Energy, Kcal/mol
PVAC
PVC
PS
PC
98
136
194
146
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References and Additional Literature
[1] Sheth, K. C., Chen, M. J. & Farris, R. J. (1995) Mat. Res. Soc. Symp. Proc. 356,
520-534.
[2] Bird, R. B., Stewart, W. E. & Lightfoot, E. N. (1960) Transport Phenomena
(John Wiley & Sons, New York).
[3] Pinkus, O. & Sternlicht, B. (1961) Theory of Hydrodynamic Lubrication
(McGraw-Hill Book Company, New York).
[4] American Society for Testing Materials, ASTM E 1142, Terminology Relating to
Thermophysical Properties.
[5] I.M. Ward, Mechanical Properties of Solid Polymers, John Wiley and Sons Inc.,,
New York 1983, Chapter 7.
Assignment of the Glass Transition, ed. R.J. Seyler, STP 1249, ASTM, Philadelphia
(1994).
Physics and Mechanics of Amorphous Polymers, J. Perez, A.A. Balkema Publ.,
Brookfield (1998).
The Physics of Glassy Polymers, eds. R.N. Haward, R.J. Young, 2nd edition, Chapman &
Hall, London (1997).
Introduction to Polymer Physics, U. Eisele, Springer Verlag, New York (1990).
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