Satas D., Tracton A.A. (ed.). Coatings Technology Handbook
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42
VENKATRAMAN
The number
M,,
is usually measured by NMR spectrometry or osmometry;
M,,,
can
be obtained via light scattering techniques, while intrinsic viscosity measurements yield
estimates of
M,,.
Size-exclusion chromatography or gel permeation chromatography (GPA)
can in principle be used to obtain all the averages mentioned above; care must be taken
to
ensure proper calibration of the column with standards that have the same molecular
structure
as
the polymer of interest.
Although there exist different definitions for the breadth of
a
distribution, we will
use the most common one involving the averages defined above:
MWD (molecular weight dispersity)
=
-
M,,.
M,,
A value of unity for this quantity defines a “narrow distribution” polymer; a value of
2
is obtained
in
condensation polymers, and higher values indicate considerable breadth of
molecular weights. A measure
of
this quantity can be obtained via GPC, or a combination
of
NMR and light-scattering techniques.
1.2 Molecular Weight Between Cross-Links
This is defined
as
the average molar mass between successive cross-link sites in a network
polymer and is denoted by the symbol
M,..
It is a measure
of
the density of cross-linking
and can be estimated from measurements of the equilibrium degree of swelling or
of
the
modulus.
1.3 Particle Size and Particle Size Distribution
In the case of latexes, many properties of the wet and dry coatings are determined by the
sizes of the latex particles. Estimations can be obtained directly through scanning electron
micrography (SEM) if a film can be made. For the suspension, however,
it
is more custom-
ary
to use light-scattering techniques (Coulter model N4, Brookhaven model DCP-1000)
or optical sedimentation techniques (Horiba CAPA-700). In either case,
it
is possible to
obtain a major portion of the particle size distribution.
2.0 PROPERTIES
OF
WET COATINGS
Described below are some of the more important properties
of
coatings that are relevant
to their ease of application, either in solution or
as
suspensions. Most wet coatings are
brushed on (as with paints) or sprayed
(as
with some epoxies used as insulation). The
solution coatings are mostly polymer based, and thus a survey
of
the rheological properties
of polymer solutions is given; in addition, some properties
of
suspensions are discussed.
2.1
Viscosity
of
Polymer Solutions
Although several theories
of
polymer solutions’ examine the dependence of viscoelastic
properties
on
molecular parameters, we shall not discuss these here. Instead, we shall
focus on some generally accepted empirical relationships. Most
of
these are covered exten-
sively by Ferry.’
2.1.1
Dependence
on
Molecular Weight
For pure polymers, the molecular weight dependence is usually expressed by the following
type
of
relationship:
STRUCTURE-PROPERTY RELATIONSHIPS
IN
POLYMERS
43
qo
=
KMs
(2)
where
qo
is the “zero-shear” viscosity, and
K
is
a
solvent- and temperature-dependent
constant. The value of the exponent
p
is determined by the molecular weight range under
consideration:
for
M M,.,
p
=
1
and for
M
>
M,.,
p
=
3.4 (3)
where
M,.
is a critical molecular weight that expresses the onset of entanglements between
molecules. The magnitude
of
M,.
is characteristic of the polymer structure; Table
1
gives
some representative numbers.
Although
M,.
signals the onset
of
topological effects on the viscosity, it is not identical
to
the molecular weight between entanglements,
M,..
(The latter quantity is estimated from
the magnitude
of
the rubbery plateau modulus.) Approximately, we have
M,.
-
2M,.
(4)
Also,
M,.
is a function
of
polymer concentration.
In
the pure polymer (denoted by super-
script zero), it attains its lowest value,
M:!;
in a solution of concentration
C,
its magnitude
varies as discussed in Section
2.1.2.
The exponent
p
assumes the values quoted in Equation
3
only if the measured
viscosity is in the so-called zero-shear-rate limit. At higher rates,
p
assumes values lower
than unity and 3.4, in the
two
regimes.
2.1.2
Concentration Dependence
of
the ViscosiQ
As mentioned in Section
2.1.1,
below a certain concentration,
C*,
entanglement effects
are not significant. This concentration is estimated from:
Mjl
c*
=
p-
M
where
M
is the molecular weight of the polymer in the coating solution. The concentration
C*
cannot be estimated from
a
plot of
qo
against concentration, however; the transition
is not sharp, but gradual.‘
No
single expression for the concentration exists below
C*;
however. in the entangled regime, the expression
qO
=
C””.‘
(5
)
works well for some This relation does not hold all the way to the pure
Table
1
Critical Molecular
Weight
of
Source Polymers
Polymer
M,.
Polyvinyl chloride 6,200
Polyethylene 3,500
Polyvinyl acetate
25,000
Polymethyl acrylate 24,000
Polystyrene 35,000
Source:
Ref.
3
44
VENKATRAMAN
polymer, where higher exponents are found.' Equation
5
also does not hold in the case
of
polymer solutions in which there are other specific attractive forces, such as in poly(n-
alkyl acrylates).'
There are two reasons for
a
reduction in viscosity of a polymer upon dilution:
(a)
the dilution effect, which causes the solution viscosity
to
be between those
of
the two
pure components, and (b)
a
decrease of viscosity due to
a
lowering of
T,y
upon dilution. The
latter is solvent-specific, and is the main reason for the apparent difficulty in establishing a
universal viscosity-concentration relationship for polymer solutions.
2.2
Viscosity
of
Suspensions
Many latex paints are suspensions in water or
in
an organic solvent. Their rheological
properties differ from those of polymer solutions in several ways. The concentration depen-
dence is of a different form, and in addition, there is a dependence
on
particle size.
Also,
at high concentrations, these suspensions tend to have structure. which usually refers to
an aggregated network. The immediate consequences
of
the existence of
a
pseudonetwork
are the phenomena of yield stress and thixotropy. We explore next the relationship
of
these quantities
to
the characteristics
of
the particles making up the suspension.
2.2.
I
Concentration
Dependence
of
the Viscosit?,
In dilute suspensions. the concentration dependence is expressed by an extension
of
the
Einstein equation:
where
q,,
is
the solvent viscosity and
+
is the volume fraction of the suspension. Equation
6
is valid for spherical particles without any interparticle interaction. Inclusion of long-
range interaction (such as volume exclusion) merely changes the coefficient
of
the
+'
term.
Of greater interest are the rheological phenomena that occur in suspensions of parti-
cles that have short-range interactions. attractive or repulsive.
In
a
comprehensive study,
Matsumoto et aLx have established the conditions for the existence of yield stresses in
suspension. Their conclusions are:
1.
For particles with repulsive interactions, no yield stresses exist.
2.
Suspensions of neutral particles. or particles with attractive forces do exhibit
3.
The magnitude
of
the yield stress increases with the concentration
of
the particles
yield stresses.
and with increasing ratio of surface area to volume.
In this study, the existence
of
the yield stress was inferred from the presence of
a
plateau in the elastic modulus
G',
at very low frequencies; the magnitude of the yield
stress was deduced from the height of the plateau modulus.
A
detailed and critical survey
of
the literature is given by Meitz.9
The other impostant rheological consequences
of
a
pseudonetwork is thixotropy,
defined elsewhere
in
this volume. The phenomenon is attributed
to
a time-dependent but
reversible breakdown of
the
network.
STRUCTURE-PROPERTY RELATIONSHIPS IN POLYMERS
45
3.0
PROPERTIES
OF DRIED FILMS
3.1
The Glass Transition Temperature
The
T,
is defined in various ways, but in a broad sense. it signals the onset of small-scale
motion in
a
polymer. It is heavily influenced by the chemical structure,
in
particular, by
the bulkiness (steric hindrance) of pendant groups. (See Van Kreleven'" for an excellent
discussion.)
The molecular weight dependence of the glass transition is fairly straight forward
and is given by:
where is the limiting value of
T,
at high molecular weights.
treatment by Bueche,'
I
the
T,
reduction by plasticizers can be calculated from:
The effect of plasticizers on
T,
is
well documented.'" According to a theoretical
T,
(solution)
=
TS,,I
+
(KT,,,,
-
7-s.,J+.>
1
+
(K
-
l)+,,
where
TV,,,
and
TV,,,
are the glass transition temperatures of the polymer and solvent, respec-
tively, and
4,
is the volume fraction of solvent. Then we write
where
aI
=
volume expansion coefficient above
T,
a,,
=
volume expansion coefficient below
T,
To calculate the
T,
of a solution, the
T,
of the solvent should be known" or estimated
as
2T,,,/3,
K
is usually taken to be
2.5.
3.2
Tensile and Shear Moduli
Both tensile and shear moduli are functions of temperature, and of time in the case of
viscoelastic polymers. We shall restrict our discussion to the temperature dependence
of
the isochronal moduli. (Since the tensile and shear moduli are related to each other through
the use of an equation involving the Poisson's ratio, the comments made here
on
the shear
modulus
G
can be extended to the tensile modulus
E,
as well.)
For semicrystalline polymers below
TV,
the modulus can be estimated from:13
G
=
G,
+
X;'(G,.
-
G,)
(9
1
where
G,,,
G,.
are the moduli for the fully amorphous and fully crystalline polymer, respec-
tively, and
X,.
is the degree
of
crystallinity.
Above
Tq,
the same equation can be used, but
G,.
far exceeds
G,c,
and Equation
9
reduces to
For amorphous polymers above
T,,
the modulus is given by the rubber elasticity expres-
sion:
46
VENKATRAMAN
PK7
G,,
=
-
M,.
where
Mj!
is the entanglement molecular weight (see
(1
1)
Section 2.1.2). This rubbely plateau
for un-cross-linked polymers is observed only at molecular weights higher than
M(!.
in analogous fashion:
"
For cross-linked amorphous polymers above
TV
(elastomers), the modulus is given
KT
G=P
M,.
(12)
where
M,.
is the molecular weight between permanent cross-linked junctions. When
M,.
exceeds
M,,,
trapped entanglements also play
a
part in determining modulus:"
where
G,.
is given by Equation
11
and$. is a probability factor for trapped entanglements.
In the case
of
network imperfections, Equation 12 is The quantity
L.
can be calculated
if
the reaction parameters for network formation are kno~n.~".'""'
3.3
Other
Properties
Several other properties of dried films influence its performance characteristics. Examples
are the coefficient of thermal expansion, ultimate mechanical properties, stress relaxation
and creep, and dielectric properties. However, correlation of these properties with structure
for polymeric films is not well established; some of the more successful attempts are
treated
in
Refs. 2 and 3.
REFERENCES
I.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
R.
B.
Bird, R. C. Armstrong,
and
0.
Hassager,
Dytmnics
of
Polymeric
Fluids,
Vol. 2. New
York: Wiley-Interscience, 1987.
J.
D. Ferry,
Viscoelastic Properties
of
Polytners.
New York: Wiley, 1980.
D. W.
Van
Krevelen,
Properties
of
Polyrners.
New
York: Elsevier, 1976.
Ref.
2,
see discussion
in
Chapter 17.
J.
W. Berge
and
J.
D.
Ferry,
J.
Colloid
Sci.,
12.
400 (1957).
G. Pezzin and
N.
Gligo,
J.
Appl.
Polym.
Sci.,
IO,
1
(1966).
Ref.
2,
p.
510.
T.
Matsumoto,
0.
Yamamoto,
and
S.
Onogi,
J.
Rheol.,
24,
279 (1980).
D. W. Meitz, Ph.D. thesis, Carnegie-Mellon
University,
December 1984.
Ref. 3, p. 383.
F.
Bueche,
Physical Properties
qf
Polymers.
New
York:
Wiley, 1962.
Ref. 3, p. 384.
Ref. 3, p. 266.
E.
M.
Valles and C. W. Macosko,
Mrrcrondecu1e.s.
12,
673 (1979).
P.
J.
Flory,
Principles
of
Polymer Chemistry.
Ithaca,
NY:
Cornell University Press, 1953,
p.
458.
M.
Gottlieb,
C.
W. Macosko, G.
S.
Benjamin,
K.
0.
Meyers, and E. W. Merill,
Mwrotnolecules,
14,
1039 (1981).
D.
S.
Pearson and
W.
W. Graessley,
Macrotnolecules,
13,
1001 (1980).
5
The Theory
of
Adhesion
When pressure-sensitive adhesive is applied
to
a smooth surface,
it
sticks immediately.
The application pressure can be very slight, not, more than the pressure due to the weight
of the tape itself. The adhesive is said to “wet” the surface, and, indeed,
if
the tape is
applied
to
clear glass and one views the attached area through the glass, it is found that
in certain areas the adhesive-glass interface looks like a liquid-glass interface. From this
one would infer that
a
pressure-sensitive adhesive, even though
it
is
a
soft, highly compliant
solid.
also
has liquidlike characteristics. Some knowledge of the interaction between liquids
and solids is beneficial to the understanding of adhesion.
1
.O
CONTACT ANGLE EQUILIBRIUM
When
a
drop of liquid is placed on
a
surface of a solid that is smooth, planar, and level,
the liquid either spreads out
to
a thin surface film, or it forms a sessile droplet on the
surface. The droplet has
a
finite angle of contact (Fig.
l).
The magnitude
of
the contact
angle depends on the force of attraction between the solid and the liquid and the surface
tension
of
the liquid. The contact angle equilibrium has received
a
great deal
of
attention,
principally because it is perhaps the simplest direct experimental approach to the thermody-
namic work of adhesion.
Many years
ago
Young’ proposed that the contact angle represents the vectorial
balance
of
three tensors, the surface tension
of
the solid in air
(y,(,),
the surface tension
of the liquid in equilibrium with the vapor
(y/,.),
and the interfacial tension between the
solid and the liquid
(ysI).
The force balance can be written
yso
=
y/v
cos
e
+
Y,/
(1)
Young’s equation has come under criticism on the grounds that the surface tension of a
solid is
ill
defined, but most surface chemists find his equation acceptable
on
theoretical
grounds.
47
40
DAHLQUIST
Figure
1
Contact angle equilibrium
of
surface
and
interfacial tensions.
The equation can be written as a force equilibrium or
as
an energy equilibrium,
because the surface tension, expressed
as
a force per unit of length, will require an energy
expenditure of the same numerical value when it acts to generate a unit area of new
surface.
Harkins and Livingston' recognized that Young's equation must be corrected when
the exposed surface of the solid carries an adsorbed film of the liquid's vapor. The
solid-"air-plus-vapor" tensor,
Y,,~.,
is less than the solid-air tensor,
Y,,(~.
Harkins and
Livingston introduced
a
term,
ne,
to indicate the reduction thus:
Y.51.
=
YW
-
(2)
and the modified Young equation becomes accordingly
y.w
=
?/l.
cos
f3
+
y,/
+
ne
(3)
The work of adhesion
(Wcl)
is the thermodynamic work necessary to separate the
liquid from the solid without performing any additional work, such
as
viscous or elastic
deformation of either the liquid or the solid. Duprt' is credited with the definition
of
the
work of adhesion. When a liquid is separated from a solid, new solid-air and liquid-air
interfaces are created, and solid-liquid interface is destroyed
(Fig.
2).
Duprt equation becomes
Combining Equations
4
and
3
yields
W,,
=
Y/\.
(1
+
cos
8)
+
ne
THE THEORY
OF
ADHESION
49
Under some circumstances
Te
is negligible, and Equation
5
simply reduces to the
original Young-DuprC equation
W,,
=
y/,.
(1
+
cos
8)
(6)
The work of adhesion can then be obtained by measuring the surface tension of the liquid
and the contact angle;
y,,(,
need not be known.
Zisman and his associates4,s at the Naval Research Laboratories have done extensive
and careful investigation of the contact angles of liquids on solids. They found that when
the surface tensions of a family of liquids were plotted against the cosines of their contact
angles
on
a given solid, the data fell, with some scatter, along a relatively straight line.
The surface tension value at which this line intersected the ordinate [i.e., where cos
8
=
1.00
(or
8
=
O")],
was designated the critical surface tension,
y,..
Some have equated
yc.
of a solid to its surface tension or surface energy, but this
was never done or condoned by Zisman. The critical surface tension of a solid is simply
the highest surface tension among the surface tensions
of
liquids that will spread on the
solid. It is, of course, a measure of the attractive force a liquid experiences when it comes
in contact with the solid.
Some have postulated that a contact angle
of
0"
is essential for the formation of a
strong adhesive bond. This is said to be requisite for wetting. Wetting, however, is a
relative term. Surely the molecules of a liquid and a solid will be
in
close contact at the
interface even though the contact angle is greater than
0".
In fact,
if
one measures the
works of adhesion between a family of liquids and a solid, one finds, as illustrated in
Figure
3,
that the work
of
adhesion can maximize at a relatively high contact angle. Thus
spreading is not requisite for good adhesion on thermodynamic grounds, but
it
may be
wA
ergs
cm
2
Contact
Angle
Figure
3
Work
of
adhesion
of
liquids on
a
surface
having
a
y<.
of
21.5
ergskm'.
50 DAHLQUIST
highly desirable in real situations for maximizing contact area and minimizing interfacial
flaws and defects.
2.0
FORCES
OF
ATTRACTION
All
atoms or molecules, when brought into sufficiently close proximity, exhibit attraction
for each other
as
a
result of asymmetric charge distributions in their electron clouds. Under
special circumstances,
if
there are fixed electrical charge distributions (e.g., in electrets)
or fixed magnetic polarization, repulsion may occur, but these are special and rare cases.
Repulsion also occurs when matter is compressed to the point that the outer electron orbits
begin to interpenetrate each other; but when electrical charges and magnetic domains are
randomly distributed, and the liquids and solids are
not
under extremely high compressive
forces, the forces of attraction prevail.
The forces take several forms. The forces due
to
the fluctuation in the electron
cloud density are universal. These forces were first recognized by Fritz London” and are
sometimes called the “London forces,” but more frequently they are called “dispersion
forces.” The fluctuation in electron density produces temporary electrical dipoles, which
give rise to universal attraction in matter. These are the forces that account for cohesion
and adhesion in aliphatic hydrocarbons. They are relatively weak atomic interactions, but
since all the atoms are involved, the cumulative effect is great. The dispersion forces
depend markedly on the distance of separation, decreasing as the seventh power of the
separation between isolated atom pairs, and
as
the third or fourth power of the separation
between planar surfaces.
The dispersion forces are common to all matter and can account for universal adhe-
sion in condensed media, but other forces can contribute substantially to adhesion in
specific cases. There may be attractive forces between permanent dipoles, where the poten-
tial energy
of
head-to-tail interaction of two dipoles having dipole moments
p1
and is
given in the lowest energy configuration,
as
where
r
is the center-to-center distance between the dipoles.
If the rotational energy is less than the thermal energy of the system, then
U,
=
Keesom potential
=
~
-
2k.7&
3kTr”
where
k
is Boltzmann’s constant
(0.0821
l.atm/mol deg) and
T
is absolute temperature
(K).
There may be dipole-induced dipoles, where the potential energy of interaction is
given by
U’
=
-
k:a2
+
F&,
r’
where
a?
and
a,
are the molecular polarizabilities.
There may be acid-base interactions7,’ across the interface which can lead
to
strong
bonding. Examples are hydrogen bonding, Lewis acid-base interactions, and Bronsted-
type acid-base interactions.
THE THEORY
OF
ADHESION
D1
Covalent bonding between adhesive and adherend,
if
achievable either by chemical
reactions or by high energy radiation, can lead to very strong bonds.
Interdiffusion, usually not achievable except between selected polymers, can
also
lead to high adhesion.
The force of attraction between planar surfaces has been derived from quantum
mechanical considerations by Casimir, P~lder,~ and Lifshitz."
Lifshitz calculated the attractive forces between nonmetallic solids at distances of
separation sufficiently large that the phase lag due to the finite velocity of electromagnetic
waves becomes a factor. He obtained the following relationship between the attractive
force and the known physical constants:
where
F
is the attractive force per unit of area,
h
is Planck's constant,
C
is the velocity
of light,
d
is the distance of separation,
e,,
is the dielectric constant, and
+(e,,)
is a multiply-
ing factor that depends on the dielectric constant as follows:
1
le,,:
0
0.025
0.10
0.25
0.50
1
+(eo):
1
0.53
0.4
1
0.37
0.35
0.35
Strictly speaking, the dielectric constant
in
this expression should be measured at
electron orbital frequency, about
10Is
Hz. However,
if
we assume handbook values of the
dielectric constant at
10"
Hz, which for nylon, polyethylene, and
polytetrafluoroethylene,
are 3.5,2.3, and
2.0,
respectively, the corresponding
+(e,,)
values are 0.37,0.36, and 0.35.
The force values then stand
in
the ratios
0.1
1
to 0.056 to 0.039. When normalized to
F
(nylon)
=
1.0,
they fall to the following ratios:
F(nylon),
=
1
.OO;
F(PE),
0.5
I;
F(PTFE), 0.35
When the
Y~.
values (dyneskm) of these three materials are similarly normalized to the
y<.
values for nylon, the values fall in remarkedly similar ratios.
Nylon PE PTFE
56
1
.oo
31
0.55
18.5
0.33
In the Lifshitz equation, the force of attraction is shown to decrease as the inverse
fourth power of the distance of separation. However, when the separation becomes so
small that the phase lag
in
the interaction no longer is significant (it is of the order of 6"
at a separation of
50
A),
the attractive force varies as the inverse third power of the distance
of separation. This has been verified experimentally," although the direct measurement
is extremely difficult (Fig.
4).
The forces existing at separations greater than
50
A
contrib-
ute very little to adhesion.
Some 30 years ago Good and Girifalco reexamined the interfacial tensions between
dissimilar liquids and developed a theory of adhesion." They found that the work of
adhesion, given by