`
`
`
`322 Influence of Processing in Propeties
`
`
`
`8 Solidification of Polymers 323
`
`—$4
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`material into a leathery or rubberlike substance. Once the material has cooled
`below the glass transition temperature, 7., the polymer becomesstiff and
`brittle. At the glass transition temperature,
`the specific volume and enthalpy
`curves experience a significant change in slope. This can be seen for
`polystyrene in the enthalpy-temperature curve shown in Fig. 8.1. With semi-
`crystalline thermoplastics, at a crystallization temperature near the melting
`temperature, the molecules start arranging themselves in small crystalline and
`amorphous regions, creating a very complicated morphology. During the
`process of crystalline structure formation, a quantum of energy, often called
`heatofcrystallization orheatoffusion, is released and must be conducted out of the
`material before the cooling process can continue. The heat of fusion is reflected
`in the shape of the enthalpy-temperature curve as shown for polyamide 6.6,
`polyethylene and polypropylene in Fig. 8.1. At the onset of crystalline growth,
`the material becomes rubbery yet notbrittle, since the amorphous regions are
`still above the glass transition temperature. As seenearlier, the glass transition
`temperature
`for
`some
`semi-crystalline
`polymers
`is
`far below room
`temperature, making them tougher than amorphous polymers. For common
`semi-crystalline polymers, the degree of crystallization can be between 30 and
`70%. This means that 30-70%of the molecules form crystals and the rest remain
`in an amorphous state. The degree of crystallization is highest for those
`materials with short molecules since they can crystallize faster and more easily.
`
`
`840
`
`630
`
`2107
`
`(J/g)
`
`0
`
`0
`
`80
`
`!
`240
`
`_— i
`160
`
` Temperature, T (°C)
`
`p=1 bar
`
`——
`320
`
`420 Enthalpy
`
`
`Figure 8.1.
`
`Enthalpyas a function of temperature for various thermoplastics.
`
`Figure 8.2 [1] shows the volumetric temperature dependence of a polymer.
`In the melt state, the chains have “empty spaces” in which molecules can move
`freely. Hence, undercooled polymer molecules canstill move as long as space
`is available. The point at which this free movement ends for a molecule or
`segment of chains is called the glass transition temperature or solidification
`point. As pointed outin Fig. 8.2, the free volume is frozen-in as well. In the case
`
`
`
`of crystallization, ideally, the volume should jump to a lower specific volume.
`However even here, small amorphous regions remain which permit a slow
`flow or material creep. This free volume reduces to nothing at absolute zero
`temperature at which heat transport can no longer occur.
`
`
`
`Tm
`Ty
`T-.
`T=0K
`Thermal expansion modelfor thermoplastic polymers.
`
`Tigure 8.2
`
`The specific volume of a polymer changes with pressure even at the glass
`transition temperature. This is demonstrated for an amorphous thermoplastic
`in Fig. 8.3 and for a semi-crystalline thermoplastic in Fig. 8.4.
`It should be noted here that the size of the frozen-in free volume depends
`on the rate at which a material is cooled; high cooling rates result in a large
`free volume.In practice this is very important. When the frozen-in free volume
`is large, the part is less brittle. On the other hand, high cooling rates lead to
`parts that are highly permeable, which may allow the diffusion of gases or
`liquids through container walls. The cooling rate is also directly related to the
`dimensional stability of the final part. The effect of high cooling rates can often
`be mitigated by heating the part to a temperature that enables the molecules to
`move freely; this will allow furthercrystallization by additional chain folding.
`This process has a great effect on the structure and properties of the crystals and
`is
`referred to as annealing.
`In general,
`this only signifies a qualitative
`improvement of polymerparts. It also affects shrinkage and warpage during
`service life of a polymer component, especially when thermally loaded.
`All these aspects have a great impact on processing. For example, when
`extruding amorphous thermoplastic profiles,
`the material can be sufficiently
`cooled inside the dic so that the extrudate has enoughrigidity to carry its own
`weight as it is pulled away from the die. Semi-crystalline polymers with low
`molecular weights have a viscosity above the melting temperature that is too
`low to be able to withstand their own weight as the extrudate exits the die.
`Temperatures below the melting temperature, T,, however cannot be used due
`to solidification inside the die. Similar problems are encountered in the
`
`MacNeil Exhibit 2117
`
`Yita v. MacNeil IP, IPR2020-01139
`Page 172
`
`MacNeil Exhibit 2117
`Yita v. MacNeil IP, IPR2020-01139
`Page 172
`
`

`

`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`324 Influence of Processing in Propeties
`
`8 Solidification of Polymers
`
`325
`
`thermoforming process in which the material must be heated to a point so that
`it can be formed into its final shape, yet has to be able to withstand its own
`weight.
`
`(ag > a1 >a2)sq
`
`
` SP OG| 1
`i—*ag2) 1
`|
`|_| Freezing line |
`Tao?
`Tgo. Tg
`Ts
`Temperature, T
`Schematic of a p-v-T diagram for amorphous thermoplastics.
`
`Figure 8.3
`
`(agg > ag1 > ag2)
`
`Melt
`
`\Po
`
`longer
`be removed to solidify the part; and since there is more shrinkage,
`packing times and larger pressures must be employed. All this implies longer
`cycle times and more shrinkage. High cooling rates during injection molding
`of
`semi-crystalline polymers will
`reduce the degree of crystallization.
`However, the amorphousstate of the polymer molecules may lead to some
`crystallization after the process, which will result in further shrinkage and
`warpage of the final part. It is quite common to follow the whole injection
`molding process in the p-v-T diagrams presented in Figs. 8.3 and 8.4, and thus
`predict how much the molded component has shrunk.
`
`8.1.2 Morphological Structure
`
`Morphology is the order or arrangement of the polymerstructure. The possible
`“order” between a molecule or molecule segment and its neighbors can vary
`from a very ordered highly crystalline polymeric structure to an amorphous
`structure (i.e., a structure in greatest disorder or random). The possible range of
`order and disorderis clearly depicted ontheleft side of Fig. 8.5. For example, a
`purely amorphous polymer
`is
`formed only by the non-crystalline or
`amorphous chain structure, whereas the semi-crystalline polymer is formed by
`a combination of all the structures represented in Fig. 8.5.
`The imageof a semi-crystalline structure as shownin the middle of Fig. 8.5,
`can be captured with an electron microscope. A macroscopic structure, shown
`in the right hand side of
`the figure, can
`be captured with an optical
`microscope. An
`optical microscope
`can capture the coarser macro-
`morphological structure such as the spherulites in semi-crystalline polymers.
`
`Specific
`
`
` Specific
`volume,V
`
`volume,V
`'
`
`
`— ia W (W) \
`Texture
`Ny rags
`ieaind
`Charafteristic
` coe
`= >
`ca. 0.01-0.02tm ie
`4
`Ue
`|
`oe
`
`/Semi-crystalline Inhomogeneous semi-
`Ss / structure
`crystalline structure
`Bevan /
`characteristic size
`
`
`Amorphous9stp 1-2 im
`be
`
`T20°
`Tmo Tm1
`Ts
`
`Temperature, T
`
`Schematic of a p-v-T diagram for semi-crystalline thermoplastics.
`
`
`Figure 8.4
`
`Semi-crystalline polymers are also at a disadvantage in the injection
`molding process. Because of the heat needed for crystallization, more heat must
`
`Figure 8.5
`
`Schematic diagram of possible molecular structure which occur in
`thermoplastic polymers.
`
`An amorphous polymer is defined as having a purely random structure.
`Howeverit is not quite clear if a “purely amorphous” polymer as such exists.
`Electron microscopic observations have shown amorphous polymers that are
`
`MacNeil Exhibit 2117
`
`Yita v. MacNeil IP, IPR2020-01139
`Page 173
`
`MacNeil Exhibit 2117
`Yita v. MacNeil IP, IPR2020-01139
`Page 173
`
`

`

`
`composedof relatively stiff chains, show a certain degree of macromolecular
`
`
`structure and order, for example, globular regions or fibrilitic structures.
`
`
`Nevertheless, these types of amorphous polymersarestill found to be optically
`
`
`isotropic. Even polymers with soft and flexible macromolecules, such as
`
`
`polyisoprene which wasfirst considered to be random, sometimes show band-
`
`
`like and globular regions. These bundle-like structures are relatively weak and
`
`
`short-lived when the material experiences stresses. The shear thinning viscosity
`
`
`effect of polymers
`sometimes
`is attributed to the breaking of
`such
`
`
`macromolecular structures.
`
`8.1.3 Crystallization
`
`
`
`8 Solidification of Polymers
`
`327
`
`
`
`
`|
`
`
`
`326 Influence of Processing in Propeties
`
`
`
`8.8 [4].
`written as follows:
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`the
`Early on, before the existence of macromolecules had been recognized,
`presence of highly crystalline structures had been suspected. Such structures
`were discovered when undercooling or when stretching cellulose and natural
`rubber. Later,
`it was found that a crystalline order also existed in synthetic
`macromolecular materials such as polyamides, polyethylenes, and polyvinyls.
`Because of the polymolecularity of macromolecular materials, a 100% degree of
`crystallization cannot be achieved. Hence, these polymers are referred to as
`semi-crystalline. It is common to assumethat the semi-crystalline structures are
`formed by small regions of alignment or crystallites connected by random or
`amorphous polymer molecules.
`With the use of electron microscopes and sophisticated optical microscopes
`the various existing crystalline structures are now well recognized. They can be
`listed as follows:
`
`* Single crystals. These can form in solutions and help in the study of
`crystal formation. Here, plate-like crystals and sometimes whiskers are
`generated.
`Spherulites. As a polymer melt solidifies, several folded chain lamellae
`spherulites form which are up to 0.1 mm in diameter. A typical example
`of a spherulitic structure is shownin Fig. 8.6 [2]. The spherulitic growth
`in a polypropylene melt is shownin Fig. 8.7 [3].
`is deformed while
`Deformed crystals.
`If a semi-crystalline polymer
`undergoing crystallization, oriented lamellae form instead of spheru-
`lites.
`
`Shish-kebab. In addition to spherulitic crystals, which are formed by
`plate- and ribbonlike structures, there are also shish-kebab, crystals
`which are formed by circular plates and whiskers. Shish-kebab structures
`are generated when the melt undergoes a shear deformation during
`solidification. A typical example of a shish-kebab crystal is shown in Fig.
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`Figure 8.7
`
`Developmentof the spherulitic structure in polypropylene. Images were
`taken at 30 s intervals.
`
`The crystallization fraction can be described by the Avrami equation [5],
`
`MacNeil Exhibit 2117
`
`Yita v. MacNeil IP, IPR2020-01139
`Page 174
`
`MacNeil Exhibit 2117
`Yita v. MacNeil IP, IPR2020-01139
`Page 174
`
`

`

`
`
`
`328
`
`Influence of Processing in Propeties
`
`
`
`8 Solidification of Polymers
`
`329
`
`
`
`
`Table 8.1 Maximum Crystalline Growth Rate and Maximum Degree of
`x(t)=1-e*"
`Crystallinity for Various Thermoplastics
`
`
`
`
`
`Polymer=—————S—S~S*S*«<SGrowthirate(min)Maximumcrystallinity(%)
`
`where Z isa molecular weight and temperature dependentcrystallization rate
`and n the Avrami exponent. However, since a polymer cannot reach 100%
`|
`Polyethylene
`>1000
`80
`crystallization the above equation should be multiplied by the maximum
`Polyamide 66
`1000
`70
`possible degree ofcrystallization, x,.
`Polyamide 6
`200
`35
`Isotactic polypropylene
`20
`63
`x{t)= x,(1 - "|
`Polyethylene terephthalate
`7
`50
`Isotactic polystyrene
`0.30
`32
`Polycarbonate
`0.01
`25
`
`(8.2)
`
`
`
`
`i) ~— Folded-chain
`
`8.1.4 Heat Transfer During Solidification
`
`Since polymer parts are generally thin, the energy equation! can be simplified
`to a one-dimensional problem. Thus, using the coordinate description shown in
`Fig. 8.9 the energy equation can be reduced to
`
`Figure 8.8 Model of the shish-kebab morphology.
`
`Soliditied layer
`
`
`(8.1)
`
`
`
`
`
`
`
`
`
`
`
`
`lamella
`
`
`
`
`
`
`
`
`lines
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`The Avrami exponent, n, ranges between 1 and 4 depending onthe typeof
`nucleation and growth. For example,
`the Avrami exponent for spherulitic
`growth fromsporadic nuclei is around 4, disclike growth 3, and rodlike growth
`2. If the growthis activated from instantaneous nuclei, the Avrami exponentis
`lowered by 1.0 for all cases. The crystalline growth rate of various polymers
`differ significantly from one to another. This is demonstrated in Table 8.1
`which shows the maximum growth rate for various thermoplastics. The
`crystalline mass fraction can be measured experimentally with a differential
`scanning calorimeter (DSC).
`A more in-depth coverage of crystallization and structure development
`during processing is given by Eder and Janeschitz-Krieg][7].
`
`_ Cooling
`~
`
`
`Figure 8.9 Schematic diagram of polymer melt inside an injection mold.
`1
`The energy equation is discussed in Chapter 3 and can be found in its complete form in Appendix A.
`
`MacNeil Exhibit 2117
`
`Yita v. MacNeil IP, IPR2020-01139
`Page 175
`
`MacNeil Exhibit 2117
`Yita v. MacNeil IP, IPR2020-01139
`Page 175
`
`

`

`(8.5)
`T=T,
`A typical temperature history for a polystyrene plate, its properties presented
`in Table 8.2 [8], is shown in Fig. 8.10. Once the material's temperature drops
`
`below the glass transition temperature, T,, it can be considered solidified. This
`
`is shown schematically in Fig. 8.11. Of importance here is the position of the
`ns ciclcteietieretihlivctrdedhaledsteedetreeAANM
`
`t=131.0s
`solidification front, X(t). Once the solidification front equals the plate's
`
`dimension L, the solidification process is complete. From Fig. 8.10 it can be
`
`shown that the rate of solidification decreases as the solidified front moves
`
`further away from the cooled surface. For amorphous thermoplastics, the well-
`
`known Neumann solution can be used to estimate the growth of the glassy or
`
`solidified layer. The Neumann solution is written as
`
`
`
`
`
`
`
`
`
`
`
`8 Solidification of Polymers
`
`t=0.125s
`EEeeeeee
`arHNIOH
`ee
`t=6.25s
`om
`
`Li
`
`ac
`
`i,
`
`“n
`5
`
`gq
`
`331
`
`:
`
`*
`
`4
`pte,
`o4
`Pottatess @S
`fos
`+
`g
`@
`J
`i 4
`a
`{ i
`/ /
`
`“gs
`t= 25
`
`yor
`
`a
`
`-
`
`‘
`
`i
`
`of
`¢
`
`“
`
`420
`
`400 -
`
`on
`= 380 -
`©
`z
`o
`a
`£ 360 -
`5
`7
`
`
`320
`340 - ry
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`330 Influence of Processing in Propetiesce
`
`I ,FT
`PCy
`as
`C,— = k—>
`
`(8.3)
`8.3
`
`Another assumption— and to reduce warpage, usually a requirement— is a
`symmetry boundary condition:
`Ho at z=0
`az
`If the sheet is cooled via forced convection or the part is inside a perfectly
`cooled mold, the final temperature of the part can be assumed to be the second
`boundary condition:
`
`(8.4)
`
`
`
`
`
`
`
`
`300 ~
`5.0
`Z (mm)
`0.0
`Temperature history of polystyrene cooled inside a 5 mm thick mold.
`
`X(t) «Jar
`
`(8.6)
`
`Figure 8.10
`
`Table 8.2
`
`Material Properties for Polystyrene
`K
`= 0.117 W/mK
`
`= 1185J/keK
`Cp
`= 1040 kg/m?
`p
`= 80°C
`Tg
`= 3.2E9 Pa
`E
`= 0.33
`Vv
`EEE
`
`Figure 8.11
`
`Schematic diagram of the cooling process of a polymerplate.
`
`MacNeil Exhibit 2117
`
`Yita v. MacNeil IP, IPR2020-01139
`Page 176
`
`MacNeil Exhibit 2117
`Yita v. MacNeil IP, IPR2020-01139
`Page 176
`
`

`

`
`
`332 Influence of Processing in Propeties
`
`8 Solidification of Polymers
`
`333
`
`
`
`
`500
`
`480
`
`460
`
`440
`
`420
`
`400
`
`380
`
`
`
`Temperature(K)
`
`—-A— t=12.5s
`
`where a is the thermal diffusivity of the polymer. It must be pointed out here
`that for the Neumann solution, the growthrate of the solidified layer is infinite
`as time goes to zero.
`The solidification process in a semi-crystalline materials is a bit more
`complicated due to the heat of fusion or heat of crystallization, nucleation rate,
`etc. When measuring the specific heat as the material crystallizes, a peak which
`represents the heat of fusion is detected (see Fig. 4.12). Figure 8.12 shows the
`calculated temperature distribution in a semi-crystalline polypropylene plate
`during cooling. The material properties used for the calculations are shown in
`Table 8.3 [8]. Here,
`the material
`that
`is below the melting temperature,
`T,,
`is
`considered
`solid2?. Experimental
`evidence [9]
`has demonstrated
`—@ (= 25.08
`that the growth rate of the crystallized layer in semi-crystalline polymersis
`_-O— t= 50.0s
`ge0
`finite. This is mainly due to thefact that at the beginning the nucleation occurs
`
`ata finite rate. Hence, the Neumann solution presented in Eq. 8.6 as well as the
`
`340+4K EF t=213s
`
`widely used Stefan condition [10], do not hold for semi-crystalline polymers.
`
`This is clearly demonstrated in Fig. 8.13 [10] which presents measuredthickness
`
`of crystallized layers as a function of time for polypropylene plates quenched
`
`at three different temperatures. For further reading on this important topic the
`
`reader is encouraged to consult the literature [11, 12].
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`Table 8.3
`
`Material Properties for Polypropylene
`K
`= 0.117 W/mK
`
`Cp, solid
`
`= 1800J/kgK
`
`Cp, melt
`
`= 2300J/kgK
`
`0
`
`Tg
`
`= 930 kg/m?
`
`= -18°C
`
`Tm
`= 186°C
`a» = 209 kJ/kg
`
`
`
`2
`
`is maximal
`[tis well-knownthat the growth of the crystalline layer in semi-crystalline polymers
`somewhat belowthe melting ternperature, ata temperature Te. The growth speed of nuclei is zero at
`the melting temperature and at the glass transition temperature.
`
`
`
`
`
`
`
`
`
`
`300
`
`
`
`
`0.0
`
`Z (mm)
`
`:
`ae
`
`Figure 8.12
`
`Temperature history of polypropylene cooled inside a 5 mm thick mold.
`
`
`
`4
`
`3
`
`z
`
`cm
`ey 2P
`
`100 9G
`=
`
`°
`
`ae
`
`;
`
`.
`/
`! i |
`4
`5
`10
`
`0
`
`Oo
`
`120 °C
`
`15
`
`Figure 8.13 Dimensionless thickness of the crystallized layers as a function of
`dimensionless time for various temperatures of the quenching surface.
`
`t’
`
`(ta/D2)
`
`MacNeil Exhibit 2117
`
`Yita v. MacNeil IP, IPR2020-01139
`Page 177
`
`MacNeil Exhibit 2117
`Yita v. MacNeil IP, IPR2020-01139
`Page 177
`
`

`

`
`
`
`
`334 Influence of Processing in Propeties
`
`
`
`8 Solidification of Polymers 335
`
`8.2
`
`Solidification of Thermosets
`
`is water. Examples of addition polymerization are
`making phenolics
`polyurethanes and epoxies.
`An example of a cross-linking reaction of a thermoset by free radical reaction
`is the co-polymerization of unsaturated polyester with styrene molecules,
`shownin Fig. 8.14. The molecules contain several carbon-carbon double bonds
`which act as cross-linking sites during curing. An example of the resulting
`network after the chemical reaction is shownin Fig. 8.15.
`
`
`
`
`
`
`
`8.2.1 Curing Reaction
`
`In a cured thermoset, the molecules are rigid, formed by short groups that are
`connected by randomly distributed links. The fully reacted or solidified
`thermosetting polymer does not react to heat as observed with thermoplastic
`polymers. A thermoset may soften somewhat upon heating and but
`then
`degrades at high temperatures. Due to the high cross-link density, a thermoset
`component behaves as an elastic material over a large range of temperatures.
`However,
`it
`is brittle with breaking strains of usually 1 to 3%, The most
`common example is phenolic, one of the most rigid thermosets, whichconsists
`of carbon atoms with large aromatic rings that impede motion, making it stiff
`and brittle. Its general structure after cross-linking is given in Figs. 3.22 and
`3.23.
`
`Polyester
`
`Figure 8.14
`
`Symbolic and schematic representations of uncured unsaturated
`polyester.
`
`Ots
`Ou
`Orch
`CHp
`CH
`eeeeeeOOHeeCOCHot
`GH,
`oH,
`GH
`f \-CH
`|
`du
`€\-cH
`
`—CH
`
`CH
`
`CH
`
`
`
`
`
`thermosets, such as phenolics, unsaturated
`The solidification process of
`
`polyesters, epoxy resins, and polyurethanes is dominated by an exothermic
`
`chemical reaction called curing reaction. A curing reaction is an irreversible
`
`process that results in a structure of molecules that are more or less cross-
`
`linked. Some thermosets cure under heat and others cure at room temperature.
`
`Thermosets that cure at room temperature are those for which the reaction
`GOO-CH2-GHp-O0C
`006
`
`starts immediately after mixing two components, where the mixing is usually
`
`cei a +n+[CHp=CH]\
`ae
`part of the process. However, even with these thermosets,
`the reaction is
`
`COO-CHy-CH,-O0G
`COO....
`O
`accelerated by the heat released during the chemical reaction, or the exotheri.
`
` ————ee————————
`
`
`In addition,
`it
`is also possible to activate cross-linking by absorption of
`Styrene
`Polyester
`
`moisture or radiation, such as ultra-violet, electron beam, and laser energy
`
`sources [13].
`
`
`In processing, thermosets are often grouped into three distinct categories,
`namely those that undergo a /ieat activated cwre, those that are dominated by a
`
`mixing activated cure, and those which are activated by the absorption of
`
`humidity or radiation. Examples of heat activated thermosets are phenolics;
`
`examples of mixing activated cure are epoxy resins and polyurethane.
`
`
`
`
`
`
`‘Thermosets can be broken down into three categories: thermosets which
`cure via condensation polymerization, those that undergo addition polymerization and
`those that cure via free radiacal polymerization.
`
`«GH-GH-COO-CHy-CHp-OOC-CH-CH
`Condensation polymerization is defined as the growth process that results
`O-te
`On
`from combining two or more monomers with reactive end-groups, and leads
`
`to by-products such as an alcohol, water, an acid, etc. A common thermoset that
`CHo
`
`polymerizes or
`solidifies via
`condensation polymerization
`is phenol
`Figure 8.15|Symbolic and schematic representations of cured unsaturated polyester.
`formaldehyde, discussed in Chapter 3. The by-product of the reaction when
`
`
`
`
`CHp
`
`:
`
`LOD
`
`MacNeil Exhibit 2117
`
`Yita v. MacNeil IP, IPR2020-01139
`Page 178
`
`MacNeil Exhibit 2117
`Yita v. MacNeil IP, IPR2020-01139
`Page 178
`
`

`

`
`
`8 Solidification of Polymers 337
`
`
`
`8.10, it is now easy to take the DSC data and find the
`With the use of Eq.
`models that describe the curing reaction.
`
`
`
`HeatflowJ/{sg)
`
`
`
`Time (minutes)
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`336 Influence of Processing in Propeties
`
`8.2.2 Cure Kinetics
`
`As discussed earlier, in processing thermosets can be grouped into two general
`categories: heat activated cure and mixing activated cure thermosets. However, no
`matter which category a thermoset belongs to,
`its curing reaction can be
`described by the reaction between two chemical groups denoted by A and B
`which link two segments of a polymer chain. The reaction can be followed by
`tracing the concentration of unreacted As or Bs, C, or C,.
`If the initial
`concentration of As and Bs is defined as C, and C,,, the degree of cure can be
`described by
`
`Ao
`
`8.7)
`
`The degree of cure or conversion, C’, equals zero when there has been no
`reaction and equals one when all As have reacted and the reaction is complete.
`However, it is impossible to monitor reacted and unreacted As and Bs during
`the curing reaction of a thermoset polymer. It
`is known though that the
`exothermic heat released during curing can be used to monitor the conversion,
`Cc’. When small samples of an unreacted thermoset polymer are placed in a
`differential scanning calorimeter (DSC), each at a different temperature, every
`sample will release the same amount of heat, Q,. This occurs because every
`cross-linking that occurs during a reaction releases a small amountof energy in
`the form of heat. For example, Fig. 8.16 [14] showsthe heat rate released during
`isothermal cure of a vinyl ester at various temperatures.
`The degree of cure can be defined by the following relation
`
`
`
`Q
`»
`@ =
`
`0,
`
`where Q is the heat released up to an arbitrary time t, and is defined by
`
`O= [Oat
`
`8.8
`
`(8.8)
`
`89)
`
`DSC data is commonly fitted to empirical models that accurately describe the
`curing reaction. Hence, the rate of cure can be described by the exotherm, @,
`andthetotal heat released during the curing reaction, Q,, as
`
`ac 2
`dt
`Q,
`
`(8.10)
`
`
`
`
`Figure 8.16 DSC scans of the isothermal curing reaction of vinyl ester at various
`temperatures.
`
`During cure, thermoset resins exhibit three distinct phases; viscous liquid,
`gel, and solid. Each of these three stages is marked by dramatic changes in the
`thermomechanical properties of the resin. The transformation of a reactive
`thermosetting liquid to a glassy solid generally involves
`two distinct
`macroscopic transitions: molecular gelation and vitrification. Molecular
`gelation is defined as the time or temperature at which covalent bonds connect
`across the resin to form a three-dimensional network which gives rise to long
`range elastic behavior in the macroscopic fluid. This point is also referred to as
`the gel point, where C’ = C,. As a thermosetting resin cures, the cross-linking
`begins to hinder molecular movement, leading to a rise in the glass transition
`temperature. Eventually, when T, nears the processing temperature, the rate of
`curing reduces significantly, eventually dominated by diffusion. At this point
`the resin has reached its vitrification point. Figure 8.17, which presents degree
`of cure as a function of time, illustrates how an epoxyresin reaches a maximum
`degree of cure at various processing temperatures. The resin processed at
`200 °C reaches 100% cure since the glass transition temperature of fully cure
`epoxy is 190 °C, less than the processing temperature. On the other hand, the
`sample processed at 180 °C reaches 97% cure and the one processed at 160 °C
`only reaches 87% cure.
`
`MacNeil Exhibit 2117
`
`Yita v. MacNeil IP, IPR2020-01139
`Page 179
`
`MacNeil Exhibit 2117
`Yita v. MacNeil IP, IPR2020-01139
`Page 179
`
`

`

`
`
`
`
`338 Influence of Processing in Propeties
`
`© 200 [°C]
`
`
`
`100
`90
`
`80
`
`70
`60
`
`50
`
`40
`
`30
`
`20
`40
`
`Degreeofcure,c[%]
`
`0
`
`20
`
`40
`
`60
`Time, t [min]
`
`9160 [°C]
`A180 [°C]
`
`80
`
`100
`
`120
`
`Figure 8.17
`
`Degree of cure as a function time for an epoxy resin measured using
`isothermal DSC.
`
`Figures 8.16 and 8.17 also illustrate how the curing reaction is accelerated as
`the processing temperature is increased. The curing reaction of thermally cured
`thermoset
`resins is not
`immediate,
`thus the blend can be stored in a
`refrigerator for a short period of time without having any significant curing
`reaction.
`
`The behavior of curing thermosetting resins can be represented with the
`generalized time-temperature-transformation (ITT) cure diagram developed
`by Enns and Gillham [16]; it can be used to relate the material properties of
`thermosets as a function of time and the processing temperature as shown in
`Fig. 8.18.
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`8 Solidification of Polymers
`
`339
`
`Ct=C1
`
`C*=Co
`
`C*=1
`
`Gel
`
`glass
`
`7
`
`Solid glass
`
`
`nk
`SacEER SS
`
`YY
`ff
`eel En aa:
`YH
`CO WL.
`—
`7
`
`
`a FIT EEE EEEEEETEES
`
`5,~Vitrification tine
`
`Tprocess
`
`Temperature
`
`+woaa
`
`4aoa
`
`tgel
`
`tg
`
`log (time)
`
`Figure 8.18 Time-temperature-transformation (TTT) diagram for a thermoset
`
`The diagram presents various lines that represent constant degrees of cure.
`The curve labeled C=C, represents the gel point and C” =1 the fully cured
`resin. Both curves have their corresponding glass transition temperatures, T,,
`and T, for the glass transition temperature of the fully cured resin andatits
`gel point, respectively. The glass transition temperature of the uncured resin,
`To, and an S-shaped curve labeled “vitrification line,” are also depicted. The
`vitrification line represents the boundary wherethe glass transition temperature
`becomes the processing temperature. Hence, to the left of the vitrification curve
`the curing process is controlled by a very slow diffusion process. The TTT-
`diagram showsan arbitrary process temperature. The material being processed
`reaches the gel point at t=t,,, and the vitrification line at r=1r,. At this point
`the material has reached a degree of cure of C, and glass transition temperature
`of the resin is equal to the processing temperature. The material continues to
`cure very slowly (diffusion controlled) until it reaches a degree of cure just
`below C,, There are also various regions labeled in the diagram. The one
`labeled “viscous liquid” is the one wherethe resin is found from the beginning
`of processing until the gel point has been reached. The flow and deformation
`that occurs during processing or shaping must occur within this region. The
`Tegion labeled “char” must be avoided during processing,
`since at high
`
`MacNeil Exhibit 2117
`
`Yita v. MacNeil IP, IPR2020-01139
`Page 180
`
`MacNeil Exhibit 2117
`Yita v. MacNeil IP, IPR2020-01139
`Page 180
`
`

`

`
`
`
`
`340 Influence of Processing in Propeties
`
`
`
`8 Solidification of Polymers 341
`
`processing temperatures the polymer will eventually undergo thermal
`degradation.
`The model that best represents the curing kinetics of thermosetting resins
`as reflected in a TTT-diagram is a diffusion modified Kamal-Sourour reaction
`model
`[17, 18, 19]. To model autocatalytic cure kinetics,
`the model can be
`applied as
`
`dc
`
`a leeecyi-cy
`where m and n are reaction orders, and k, and k, are constants defined by
`
`»)
`
`(Ty) = Tyo AC”
`Austsy
`T.=T +
`fs 1-1- AC
`
`8.1
`6.17)
`
`is the
`where T,, is the glass transition temperature of the uncured resin, T,,
`glass transition temperature of the fully reacted network, and A is a structure
`dependent parameter theoretically equated to
`
`AC
`A= Te
`pl
`
`(8.18)
`
`The values of AC,, and AC,, are the differences in the heat capacity between
`the glassy and rubbery state for the uncured resin and the fully cured network,
`respectively. However the parameter A can also be usedasfitting parameter.
`Mixing activated cure materials such as polyurethanes will instantly start
`releasing exothermic heat after the mixture of its two components has occurred.
`The proposed Castro-Macosko curing model accurately fits this behavior andis
`written as [15]
`
`ac = ke*(1-CY
`
`(8.19)
`
`(11)
`
`(8.12)
`
`(8.13)
`
`(8.14)
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`the instantaneous glass transition temperature during cure.
`is
`where T,
`
`Equation (8.12) showsthat the overall rate constant is governed at one extreme
`
`by the Arrhenius rate constant when k, >>k*, which is the case prior to
`
`
`rfa
`vitrification, and at
`the other extreme by the diffusion rate constant when
`(8.20)
`pc, a k mc + pQ
`
`k, << ki, which is the case well after vitrification. For a system exhibiting a
`
`unique one-to-one relationship between the glass transition temperature and
`
`Assuming the material
`is confined between two mold halves at equal
`conversion, DiBenedetto’s equation [20] is one of the easiest approaches for
`
`temperatures, the use of a symmetric boundary condition at the center of the
`stoichiometric ratios to express this relationship using only a single parameter
`
`partis valid:
`as
`
`oe
`k
`Ke
`ky
`4
`Here, k are Arrhenius overall rate constants defined by
`yA
`ki =ae
`
`and
`
`ki Jae?
`
`
`
`where a, and a, are fitting parameters, E, and £,, activation energies and R
`the ideal gas constant. The constant k, in Eq. 8.12 is the diffusion rate constant
`defined as
`
`kp =Gye
`
`Abn
`Hips’
`“"e”?
`
`(8.15)
`
`where a, and b are adjustable parameters, E, is the activation energy of the
`diffusion process, and f is the equilibrium fractional free volume given by
`
`f = 0.00048(7 - 7, ) + 0.025
`
`(8.16)
`
`8.2.3 Heat Transfer During Cure
`
`A well-known problem in thick section components is that the thermal and
`curing gradients become more complicated and difficult to analyze since the
`temperature and curing behavior of the part is highly dependent on both the
`mold temperature and part geometry [21, 22]. A thicker part will result in
`higher temperatures and a more complex cure distribution during processing.
`This phenomenon becomes a major concern during the manufacture of thick
`components since high temperatures may lead to thermal degradation. A
`relatively easy way to check temperatures that arise during molding and
`curing or demolding times is desired. For example, a one-dimensional form of
`the energy equation that includes the exothermic energy generated during
`curing can be solved:
`
`MacNeil Exhibit 2117
`
`Yita v. MacNeil IP, IPR2020-01139
`Page 181
`
`MacNeil Exhibit 2117
`Yita v. MacNeil IP, IPR2020-01139
`Page 181
`
`

`

`
`
`}
`
`|
`|
`1]
`
`
`
`
`
`
`
`
`.
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`|
`
`
`
`342 Influence of Processing in Propeties
`
`d
`
`of
`“=0 at z=0
`Oo
`
`T=T,
`m
`
`(8.21)
`
`(8.22)
`
`at the mold wall.
`With the useof the finite difference technique and a six constant model that
`represents dC”/dt, Barone and Caulk [23] solved Eqs. 8.20-8.22 for the curing of
`sheet molding compound (SMC). The SMC was composed of an unsaturated
`polyester resin with 40.7% calcium carbonate and 30% glass fiber by weight.
`Figures 8.19 and 8.20 show typical
`temperature
`and degree of cure
`distributions, respectively, during the solidification of a 10 mm thick part as
`computed by Barone and Caulk. In Fig. 8.