`
`!!.! Dielectric Behavior
`
`$%,
`
`U
`Ic
`
`Ueff
`
`NB = I** Ueff*sin(cid:160)
`
`I* * sin(cid:160)
`
`I*
`
`I* * cos(cid:160)
`
`(cid:160)
`
`(cid:98)
`
`NW = I*
`
` * Ueff*cos(cid:160)
`
`f
`
`Ir
`Figure 11.8  Current-voltage diagram or power-indicator diagram of electric alternating
`currents
`
`a)
`
`U
`
`I
`
`(cid:160) = 90°
`
`Period = 360°
`
`a)
`
`U
`
`t
`
`(cid:160)
`(cid:98)
`(cid:160) = 90°
`
`I'
`
`I
`Period = 360°
`
`t
`
`Figure 11.9 Current and voltage in a condenser. I ≡ current, U ≡ voltage, t ≡ time a) Without
`dielectric losses (ideal condition), current and voltage are displaced by the phase angle
`°90 or /π 2 ; b) With dielectric loss, the current curve I ´ is delayed by the loss angle δ
`ϕ=
`
`If the condenser has losses, when tanδ > 0, a resistive current Ir is formed leading
`to a heating energy rate in the dielectric of
`= 1
`E
`UI
`tanδ
`(11.27)
`h
`eff
`2
`where Ieff represents the total current or the magnitude of the vector in Fig. 11.8.
`Using Eq. 11.25 for capacitance leads to
`*
`’
`’
`’’
`
`C
`
`=
`
`C
`
`−
`
`=
`
`C
`
`−
`
`iC
`
`1ω
`
`(11.28)
`
`R
`where C* is the complex capacitance, with C’ as the real component defined by
`
`Ad
`
`’
`
`C
`
`C
`
`’’
`
`=
`
`’
`= εε0
`
`r
`and C’’ as the imaginary component described by
`=1
`εε
`’’
`r
`0ω
`R
`Using the relationship in Eq. 11.5 we can write
`−(
`) =
`*
`’
`*
`C
`C
`C
`ε iε
`ε
`=
`’’
`r
`r
`r
`0
`0
`
`(11.29)
`
`(11.30)
`
`(11.31)
`
`Ad
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 515
`
`

`

`$%-
`
` 11 Electrical Properties of Polymers
`
`where εr* is called the complex dielectric coefficient. According to Eqs. 11.25 and
`11.31, the phase angle difference or dielectric dissipation factor can be defined by
`=I
`’
`’
`I
`If we furthermore consider that electric conductivity is determined by
`σ = 1
`(11.33)
`
`R
`then the imaginary component of the complex dielectric coefficient can be rewritten
`as
`
`εε
`
`r r’
`
`r c
`
`tanδ
`
`=
`
`dA
`
`(11.32)
`
`=
`
`ε
`’
`r
`
`tan
`
`δ
`
`
`
`ε σ
`=
`’’
`r
`ωε
`0
`Typical ranges for the dielectric dissipation factor of various polymer groups are
`shown in Table 11.2. Figures 11.10 [1] and 11.11 [1] present the dissipation factor
`tan δ as a function of temperature and frequency, respectively.
`
`(11.34)
`
`Table 11.2 Dielectric Dissipation Factor (tan δ) for Various Polymers
`tan δ
`Material
`Non-polar polymers (PS, PE, PTFE)
`< 0.0005
`Polar polymers (PVC and others)
`0.001 – 0.02
`Thermoset resins filled with glass, paper, cellulose
`0.02 – 0.5
`
`PA66
`PVC+40% TCP
`
`PVC
`
`PMMA
`
`PET
`
`PI
`B-Glass
`
`PTFE
`
`PE-HD
`
`PSU
`
`PS
`
`120
`
`°C
`
`160
`
`PF 31.5
`
`PF HP
`
`PC
`
`101
`
`tan (cid:98)
`
`102
`
`103
`
`104
`
`Dissipation factor
`
`105
`
`0
`
`40
`
`80
`Temperature
`Figure 11.10 Dielectric dissipation factor as a function of temperature for various polymers
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 516
`
`

`

`!!.! Dielectric Behavior
`
`$%%
`
`PF
`
`PA66
`
`PCTFE
`
`PVC
`
`PC
`
`B-Glass
`PS
`
`PE-HD
`
`
`
`tan (cid:98)
`
`PMMA
`
`101
`
`102
`
`103
`
`(cid:98)(cid:9)
`
`Dissipation factor
`
`Dissipation factor (tan
`104
`
`106
`Frequency
`Figure 11.11 Dielectric dissipation factor as a function of frequency for various polymers
`
`105
`100
`
`102
`
`104
`
`108
`
`1010
`
`Hz
`
`1012
`
`11.1.4 Implications of Electrical and Thermal Loss in a Dielectric
`
`The electric losses through wire insulation running high frequency currents must
`be kept as small as possible. Insulators are encountered in transmission lines or in
`high-frequency fields such as the housings of radar antennas. Hence, we would
`select materials that exhibit low electrical losses for these types of applications.
`On the other hand, in some cases we want to generate heat at high frequencies.
`Heat sealing of polar polymers at high frequencies is an important technique used
`in the manufacturing of so" PVC sheets, such as the ones encountered in auto-
`mobile vinyl seat covers.
`To assess whether a material is suitable for either application the loss properties of
`the material must be determined and the actual electrical loss calculated. To do
`this, we can rewrite Eq. 11.27 as
`
`E
`
`h = 2ω δtan
`U C
`
`or as
`
`(11.35)
`
`(11.36)
`
`C
`fU d
`E
`δ
`π
`εε
`= 2
`’ tan
`2 2
`r
`h
`0
`0
`The factor that is dependent on the material and indicates the loss is the loss factor
`r’ tan , called εr’’ in Eq. 11.34. As a rule, the following should be used:
`ε
`δ
`−10 3 for high-frequency insulation applications, and
`ε
`δ
`r’ tan <
`−10 2 cfor heating applications.
`ε
`δ
`r’ tan >
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 517
`
`

`

`*&&
`
` 11 Electrical Properties of Polymers
`
`In fact, polyethylene and polystyrene are perfectly suitable as insulators in high-
`frequency applications. To measure the necessary properties of the dielectric, the
`standard DIN 53 483 and ASTM D 150 tests are recommended.
`
` (cid:132) 11.2 Electric Conductivity
`
`11.2.1 Electric Resistance
`
`The current flow resistance, R, in a plate-shaped sample in a direct voltage field is
`defined by Ohm’s law as
`R U
`=
`I
`
`(11.37)
`
`or by
`
`dA
`
`R
`
`σ
`
`(11.39)
`
`= 1
`(11.38)
`σ
`where σ is known as the conductivity and d and A are the sample’s thickness and
`surface area, respectively. The resistance is o"en described as the inverse of the
`conductance, G,
`= 1
`R G
`and the conductivity as the inverse of the specific resistance, ρ,
`= 1
`(11.40)
`
`ρ
`The simple relationship found in Eq. 11.37–39 is seldom encountered because the
`voltage, U, is rarely steady and usually varies in cyclic fashion between 10-1 to 1011
`Hz [3].
`Current flow resistance is called volume conductivity and is measured one minute
`a"er direct voltage has been applied using the DIN 53 482 standard test. The time
`definition is necessary, because the resistance decreases with polarization. For
`some polymers we still do not know the final values of resistance. However, this has
`no practical impact, because we only need relative values for comparison. Figure
`11.12 compares the specific resistance, ρ, of various polymers and shows its
`dependence on temperature. Here, we can see that similar to other polymer proper-
`ties, such as the relaxation modulus, the specific resistance not only decreases with
`time but also with temperature.
`The surface of polymer parts o"en shows different electric direct-current resist-
`ance values than their volume. The main cause of this phenomenon is surface
`contamination (e. g., dust and moisture). We therefore have to measure the surface
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 518
`
`

`

`
`
`!!.$ Electric Conductivity
`
`*&’
`
`resistance using a different technique. One common test is DIN 53 482, which uses
`a contacting sample. Another test o"en used to measure surface resistance is DIN
`53 480. With this technique, the surface resistance is tested between electrodes
`placed on the surface. During the test, a saline solution is dripped on the elec-
`trodes causing the surface to become conductive, thus heating up the surface and
`causing the water to evaporate. This leads not only to an increased artificial con-
`tamination but also to the decomposition of the polymer surface. If during this
`process conductive derivatives such as carbon form, the conductivity quickly
`increases to eventually create a short circuit. Polymers that develop only small
`traces of conductive derivatives are considered resistant. Such polymers are poly-
`ethylene, fluoropolymers, and melamines.
`
`Polyethylene
`
`Poly vinylchloride
`
`Cellulose acetate
`PE type 31 resin
`
`Polyacrylnitrile
`Organic semiconductor
`
`Pressed graphite
`
`Constantan
`Pt
`
`-200
`
`0
`200
`Temperature, T
`
`Cu
`ºC
`
`400
`
`1020
`Ohm*m
`
`1016
`
`1012
`
`108
`
`104
`
`1
`
`10-4
`
`Specific electric resistance
`
`Figure 11.12  Specific electric resistance of polymers and metals as a function
`of temperature
`
`11.2.2 Physical Causes of Volume Conductivity
`
`Polymers with a homopolar atomic bond, which leads to pairing of electrons, do not
`have free electrons and are not considered to be conductive. Conductive polymers
`in contrast, allow for movement of electrons along the molecular cluster, because
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 519
`
`

`

`*&(
`
` 11 Electrical Properties of Polymers
`
`they are polymer salts. The classification of polymer families and a comparison to
`other conductive materials is given in Fig. 11.13.
`Potential uses of electric conductive polymers in electrical engineering include
`flexible electric conductors of low density, strip heaters, anti-static equipment,
`high-frequency shields, and housings. In semi-conductor engineering, some appli-
`cations include semi-conductor devices (Schottky-Barriers) and solar cells. In elec-
`trochemistry, applications include batteries with high energy and power density,
`electrodes for electrochemical processes, and electrochrome instruments.
`Because of their structure, polymers cannot be expected to conduct ions. Yet the
`extremely weak electric conductivity of polymers at room temperature and the fast
`decrease of conductivity with increasing temperatures is an indication that ions do
`move. They move because engineering polymers always contain a certain amount
`of added low-molecular constituents that act as moveable charge carriers. This is a
`diffusion process that acts in field direction and across the field. The ions “jump”
`from potential hole to potential hole as activated by higher temperatures (Fig.
`11.12). At the same time, the lower density speeds up this diffusion process. The
`strong decrease of specific resistance with the absorption of moisture is caused by
`ion conductivity.
`
`Metal
`
`Semi-conductors
`
`Silver, copper
`Iron
`Bismuth, mercury
`
`In, Sb
`
`Germanium
`
`Insulators
`
`Silicon
`
`Glass
`DNA
`Diamond
`Sulfur
`Quartz
`
`S cm-1
`106
`104
`102
`10
`10-2
`10-4
`10-6
`10-8
`10-10
`10-12
`10-14
`10-16
`10-18
`(cid:49)-1
`Figure 11.13  Electric conductivity of polyacetylene (trans-(CH)x) in comparison to other
`materials
`
`Polyacetylene
`
`(SN)X
`TTF (cid:37)TCNQ
`NMP (cid:37)TCNQ
`KCP
`
`Trans-(CH)x
`
`Molecular
`
`crystals(cid:35)
`
`cm-1
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 520
`
`

`

`
`
`!!.$ Electric Conductivity
`
`*&)
`
`M2
`M3
`M1
`M= Median contact points per particle
`200
`300
`100
`
`T= 25°C
`
`T= 75°C
`
`T= 200°C
`
`1012
`
`Ohm
`
`108
`
`106
`
`104
`
`102
`
`Resistance, R
`
`100
`
`0
`
`10
`
`20
`
`30
`
`Vol %
`
`50
`
`V2
`V3
`Iron particle content
`Figure 11.14 Resistance R of a polymer filled with metal powder (iron)
`
`V1
`
`Conducting polymers are useful for certain purposes. When we insulate high-
`energy cables, for example, as a first transition layer we use a polyethylene filled
`with conductive filler particles such as carbon black. Figure 11.14 demonstrates
`the relationship between filler content and resistance. When contact tracks
`develop, resistance drops spontaneously. The number of inter-particle contacts,
`M, determines the resistance of a composite. At M1 or M = 1 there is one contact
`per particle. At this point, the resistance starts dropping. When two contacts per
`particle exist, practically all particles participate in setting up contact and the
`resistance levels off. The sudden drop in the resistance curve indicates why it is
`difficult to obtain a medium specific resistance by filling a polymer.
`Figure 11.15 [4] presents the resistance of epoxy resins filled with metal flakes
`or powder. The figure shows how the critical volume concentration for the epoxy
`systems filled with copper or nickel flakes is about 7 % concentration of filler, and
`the critical volume concentration for the epoxy filled with steel powder is approx.
`15 %.
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 521
`
`

`

`*&$
`
` 11 Electrical Properties of Polymers
`
`Steel particles
`Nickel flakes
`Copper flakes
`
`1012
`
`Ohm
`
`106
`
`104
`
`102
`
`Resistance, R
`
`0
`
`0
`
`10
`
`20
`Filler, (cid:113)
`Figure 11.15 Resistance of epoxy systems filled with metal flakes or powder
`
`30
`
`%
`
`40
`
` (cid:132) 11.3 Application Problems
`
`11.3.1 Electric Breakdown
`
`Because the electric breakdown of insulation may lead to failure of an electric com-
`ponent or may endanger people handling the component, it must be prevented.
`Hence, we have to know the critical load of the insulating material to design the
`insulation for long continuous use with the appropriate degree of confidence. One
`of the standard tests used to generate this important material property data for
`plate or block-shaped specimens is DIN 53 481. This test neglects the effect of
`material structure and of processing conditions. From the properties already
`described, we know that the electric breakdown resistance or dielectric strength
`must depend on time, temperature, material condition, load application rate, and
`frequency. It is also dependent on electrode shape and sample thickness. In prac-
`tice, however, it is very important that the upper limits measured on the experi-
`mental specimens in the laboratory are never reached. The rule of thumb is to use
`long-term load values of only 10 % of the short-term laboratory data. Experimental
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 522
`
`

`

`
`
`!!.% Application Problems
`
`*&*
`
`evidence shows that the dielectric strength decreases as soon as crazes form in a
`specimen under strain and continues to decrease with increasing strain. This is
`demonstrated in Fig 11.16 [5].
`
`PP(T= 40°C)
`Ed= f((cid:161)mech )
`
`U= 100 kV
`
`min
`
`230
`
`kV/mm
`
`210
`
`190
`
`170
`
`150
`
`130
`
`Dielectric strength, Ed
`
`6
`Strain, (cid:161)
`Figure 11.16 Drop of the dielectric strength of PP films with increasing strain
`
`8
`
`%
`
`12
`
`0
`
`2
`
`4
`
`On the other hand, Fig. 11.17 [5] demonstrates how the dielectric dissipation factor,
`tan δ, rises with strain. Hence, one can easily determine the beginning of the vis-
`coelastic region (begin of crazing) by noting the starting point of the change in
`tan δ. It is also known that amorphous polymers act more favorably to electric
`breakdown resistance than partly crystalline polymers. Semi-crystalline polymers
`are more susceptible to electric breakdown as a result of breakdown along inter-
`spherulitic boundaries as shown in Fig. 11.18 [6]. Long-term breakdown of semi-
`crystalline polymers is either linked to “treeing”, as shown in Fig. 11.19, or occurs
`as a heat breakdown, burning a hole into the insulation, such as the one in Fig.
`11.18. In general, with rising temperature and frequency, the dielectric strength
`continuously decreases.
`Insulation materials – mostly LDPE – are especially pure and contain voltage
`stabilizers. These stabilizers are low-molecular cyclic aromatic hydrocarbons.
`Presumably, they diffuse into small imperfections or failures, fill the empty space,
`and thereby protect the material from breakdown.
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 523
`
`

`

`*&+
`
` 11 Electrical Properties of Polymers
`
`Table 11.3 [7] provides dielectric strength and resistivity data for selected polymeric
`materials.
`
`PP
`tan (cid:98)=f((cid:161)mech,E)
`
` kV
`E= 30
` mm
`
`24
`
`20
`
`81
`
`6
`12
`
`4
`
`First registered mechanical damage
`
`3.0
`tan (cid:98)
`
`2.6
`
`2.2
`
`1.8
`
`1.4
`
`1.0
`
`Dielectric dissipation factor
`
`4
`Strain, (cid:161)
`Figure 11.17 Increase of dielectric dissipation with increased strain in PP foils
`
`6
`
`%
`
`0
`
`2
`
`Figure 11.18 Breakdown channel around a polypropylene spherulitic boundary
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 524
`
`

`

`
`
`!!.% Application Problems
`
`*&,
`
`Figure 11.19 Breakdown channel in a structure-less, finely crystalline zone of poly-propylene
`
`Table 11.3 Dielectric Strength and Resistivity for Selected Polymers
`Polymer
`Dielectric strength
`Resistivity
`(MV/m)
`(Ohm-m)
`25
`1014
`20
`1013
`20
`1013
`11
`1013
`11
`109
`10
`109
`16
`1013
`22
`1015
` 8
`1013
`15
`1012
`19
`1014
`17
`1013
`50
`1014
`12
`109
`23
`1015
`28
`1015
`20
`1014
`27
`1014
`22
`1015
`45
`1016
`14
`1012
`30
`1011
`25
`1014
`
`ABS
`Acetal (homopolymer)
`Acetal (copolymer) acrylic
`Acrylic
`Cellulose acetate
`CAB
`Epoxy
`Modified PPO
`Polyamide 66
`Polyamide 66 + 30 % GF
`PEEK
`PET
`PET + 36 % GF
`Phenolic (mineral filled)
`Polycarbonate
`Polypropylene
`Polystyrene
`LDPE
`HDPE
`PTFE
`uPVC
`pPVC
`SAN
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 525
`
`

`

`*&-
`
` 11 Electrical Properties of Polymers
`
`11.3.2 Electrostatic Charge
`
`An electrostatic charge is o"en a result of the excellent insulation properties of
`polymers – the very high surface resistance and current-flow resistance. Because
`polymers are bad conductors, the charge displacement of rubbing bodies, which
`develops with mechanical friction, cannot equalize. This charge displacement
`results from a surplus of electrons on one surface and a lack of electrons on the
`other. Electrons are charged positively or negatively up to hundreds of volts. They
`release their surface charge only when they touch another conductive body or a
`body that is inversely charged. O"en the discharge occurs without contact, as the
`charge arches through the air to the close-by conductive or inversely charged body,
`as demonstrated in Fig. 11.20. The currents of these breakdowns are low. For
`example, there is no danger when a person suffers an electric shock caused by a
`charge from friction of synthetic carpets or vinyls. There is danger of explosion,
`though, when the sparks ignite flammable liquids or gases.
`
`Discharges
`
`Flow of charge
`through polymer part
`Figure 11.20 Electrostatic charges in polymers
`
`As the current-flow resistance of air is generally about 109 Ωcm, charges and flash-
`overs only occur, if the polymer has a current-flow resistance of > 109 to 1010 Ωcm.
`Another effect of electrostatic charges is that they attract dust particles on polymer
`surfaces.
`Electrostatic charges can be reduced or prevented by the following means:
` (cid:131) Reduce current-flow resistance to values of < 109 Ωcm, for example by using con-
`ductive fillers such as graphite.
` (cid:131) Make the surfaces conductive by using hygroscopic fillers that are incompatible
`with the polymer and surface. It can also be achieved by mixing-in hygroscopic
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 526
`
`

`

`
`
`!!.% Application Problems
`
`*&%
`
`materials such as strong soap solutions. In both cases, the water absorbed from
`the air acts as a conductive layer. It should be pointed out that this treatment
`loses its effect over time. Especially, the rubbing-in of hygroscopic materials has
`to be repeated over time.
` (cid:131) Reduce air resistance by ionization through discharge or radioactive radiation.
`
`11.3.3 Electrets
`
`An electret is a solid dielectric body that exhibits permanent dielectric polari-
`zation. Some polymers can be used to manufacture electrets by solidifying them
`under the influence of an electric field, by bombarding them with electrons, or
`sometimes through mechanical forming processes.
`Applications include films for condensers (polyester, polycarbonate, or fluoropoly-
`mers).
`
`11.3.4 Electromagnetic Interference Shielding (EMI Shielding)
`
`Electric fields surge through polymers as shown schematically in Fig. 11.20. Because
`we always have to deal with the influence of interference fields, signal-sensitive
`equipment such as computers cannot operate in polymer housings. Such housings
`must therefore have the function of Faradayic shields. Preferably, a multilayered
`structure is used – the simplest solution is to use one metallic layer. Figure 11.21
`classifies several materials on a scale of resistances. We need at least 102 Ωcm for a
`material to fulfill the shielding purpose. The best protective properties are achieved
`with carbon fibers or nitrate coated carbon fibers used as fillers. The shielding prop-
`erties are determined using the standard ASTM ES 7-83 test.
`
`Polymers
`
`4
`
`6
`
`8
`
`12
`
`log ((cid:49)*cm)
`
`16
`
`Metal-Plastics
`
`Metal
`
`-6
`
`-4
`-2
`0
`2
`Electric resistance
`
`a)
`
`Al
`
`Ceramic
`
`Glass
`
`Metal-Plastics
`
`Polymers
`
`Insulators
`
`b)
`
`-5
`
`-4
`
`log [cal/(cm*s)]
`
`0
`
`-2
`-3
`Thermal resistance
`Figure 11.21 Comparison of conductive polymers with other materials: a) Electric resistance
`ρ of metal-plastics compared to resistance of metals and polymers b) Thermal resistance λ of
`metal-plastics compared to other materials
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 527
`
`

`

`*’&
`
` 11 Electrical Properties of Polymers
`
` (cid:132) 11.4 Magnetic Properties
`
`External magnetic fields have an impact on substances that are exposed to them
`because the external field interacts with the internal fields of electrons and atomic
`nuclei.
`
`11.4.1 Magnetizability
`
`Pure polymers are diamagnetic; that is, the external magnetic field induces mag-
`netic moments. However, permanent magnetic moments, which are induced on
`ferromagnetic or paramagnetic substances, do not exist in polymers. This magnet-
`izability M of a substance in a magnetic field with a field intensity H is computed
`with the magnetic susceptibility, χ as
`(11.41)
`H= χ
`M
`The susceptibility of pure polymers as diamagnetic substances has a very small and
`negative value. However, in some cases, we make use of the fact that fillers can
`alter the magnetic character of a polymer completely. The magnetic properties of
`polymers are o"en changed using magnetic fillers. Well-known applications are
`injection molded or extruded magnets or magnetic profiles, and all forms of elec-
`tronic storage devices.
`
`11.4.2 Magnetic Resonance
`
`Magnetic resonance occurs when a substance, in a permanent magnetic field,
`absorbs energy from an oscillating magnetic field. This absorption develops as a
`result of small paramagnetic molecular particles stimulated to vibration. We use
`this phenomenon to a great extent to clarify structures in physical chemistry.
`Methods to achieve this include electron spinning resonance (ESR) and, above all,
`nuclear magnetic resonance (NMR) spectroscopy.
`Electron spinning resonance becomes noticeable when the field intensity of a static
`magnetic field is altered and the microwaves in a high frequency alternating field
`are absorbed. Because we can only detect unpaired electrons using this method,
`we use it to determine radical molecule groups.
`When atoms have an odd number of nuclei, protons and neutrons, the magnetic
`fields caused by self-motivated spin cannot equalize. The alignment of nuclear
`spins in an external magnetic field leads to a magnetization vector that can be
`measured macroscopically as is schematically demonstrated in Fig. 11.22. This
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 528
`
`

`

`
`
`References
`
`*’’
`
`method is of great importance for the polymer physicist to learn more about mole-
`cular structures.
`
`2
`
`5
`
`Nuclear resonance
`signal
`
`3
`
`4
`
`1
`
`1
`
`Figure 11.22 Schematic of the operating method of a nuclear spin tomograph: 1) magnet
`producing a high, steady magnetic field; 2) radio wave generator; 3) high-frequency field,
`produced by 2, when switch 5 is in upper position; 4) processing nucleus, simulated by high
`frequency field; 5) switch; in this position the decrease of relaxation of the nucleus’ vibrations
`is measured
`
` (cid:132) References
`
`[1]
`
`[3]
`
`[4]
`
`Domininghaus, H., Elsner, P., Eyerer, P., and Hirth, T., Kunststoffe – Eigenscha#en
`und Anwendungen, Springer, Munich, (2012).
`[2] Menges, G., Haberstroh, E., Michaeli, W., and Schmachtenberg, E., Menges Werk-
`stoffkunde Kunststoffe, Hanser Publishers, Munich, (2011).
`Baer, E., Engineering Design for Plastics, Robert E. Krieger Publishing Company,
`(1975).
`Reboul, J.-P., Thermoplastic Polymer Additives, Chapter 6, J. T. Lutz, Jr., Ed., Marcel
`Dekker, Inc., New York, (1989).
`Berg, H., Ph.D Thesis, IKV, RWTH-Aachen, Germany, (1976).
`[5]
`[6] Wagner, H., Internal report, AEG, Kassel, Germany, (1974).
`Crawford, R. J., Plastics Engineering, 2nd ed., Pergamon Press, (1987).
`[7]
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 529
`
`

`

`
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 530
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 530
`
`

`

` 12
`
`Optical Properties
`of Polymers
`
`Because some polymers have excellent optical properties and are easy to mold and
`form into any shape, they are o!en used to replace transparent materials such as
`inorganic glass. Polymers have been introduced into a variety of applications, such
`as automotive headlights, signal light covers, optical fibers, fashion jewelry, chan-
`deliers, toys, and home appliances. Organic materials, such as polymers, are also
`an excellent choice for high-impact applications where inorganic materials, such
`as glass, would easily shatter. However, due to the difficulties encountered in main-
`taining dimensional stability, they are not suitable for precision optical applica-
`tions. Other drawbacks include lower scratch resistance, when compared to
`inorganic glasses, making them still impractical for applications such as automo-
`tive windshields.
`In this section, we will discuss basic optical properties, which include the index of
`refraction, birefringence, transparency, transmittance, gloss, color, and behavior of
`polymers in the infrared spectrum.
`
` (cid:132) 12.1 Index of Refraction
`
`cv
`
`n
`
`As rays of light pass through one material into another, the rays are bent due to the
`change in the speed of light from one media to the other. The fundamental material
`property that controls the bending of the light rays is the index of refraction, n. The
`index of refraction for a specific material is defined as the ratio between the speed
`of light in a vacuum to the speed of light through the material under consideration
`=
`(12.1)
`where c and v are the speeds of light through a vacuum and transparent media,
`respectively. In more practical terms, the refractive index can also be computed as
`a function of the angle of incidence, θi, and the angle of refraction, θr, as follows:
`= sin
`n
`sin
`
`(12.2)
`
`
`
`θ θ
`
`i r
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 531
`
`

`

`$%&
`
` 12 Optical Properties of Polymers
`
`where θi and θr are defi ned in Fig. 12.1.
`
`Angle of
`incidence
`
`(cid:101)i
`
`(cid:101)r
`
`Angle of
`refraction
`
` 
`
` Figure 12.1 Schematic of light
`refraction
`
`The index of refraction for organic plastic materials can be measured using the
`standard ASTM D 542 test. It is important to mention that the index of refraction is
`dependent on the wavelength of the light under which it is being measured. Figure
`12.2 shows plots of the refractive index for various organic and inorganic materials
`as a function of wavelength. One of the signifi cant points of this plot is that acrylic
`materials and polystyrene have similar refractive properties as inorganic glasses.
`
`Glass
`
`Polystyrene
`
`Quartz
`
`Acrylic
`
`1.75
`
`1.70
`
`1.65
`
`1.60
`
`1.55
`
`1.50
`
`Index of refraction, n
`
`1.45
`200
`
`600
`400
`Wavelength, (cid:104)
`
`nm 800   Figure 12.2 Index of refraction
`as a function of wavelength for various
`materials
`
`An important quantity that can be deduced from the light’s wavelength dependence
`on the refractive index is the dispersion, D, which is defi ned by
`D dn
`=
`λ
`d
`
`(12.3)
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 532
`
`

`

`
`
`!".! Index of Refraction
`
`$%$
`
`01234567
`
`104 (
`dn
`d(cid:104)
`
`Dispersion, D
`
`Polystyrene
`
`Glass
`
`Acrylic
`
`Quartz
`
`200
`
`400
`
`600
`Wavelength, (cid:104)
`
`nm
`
`800
`
` 
`
` Figure 12.3 Dispersion as a
`function of wavelength for various
`materials
`
`Figure 12.3 shows plots of dispersion as a function of wavelength for the same
`materials shown in Fig. 12.2. The plots show that polystyrene and glass have a
`high dispersion in the ultra-violet light domain.
`It is also important to mention that since the index of refraction is a function of
`density, it is indirectly affected by temperature. Figure 12.4 shows how the refrac-
`tive index of PMMA changes with temperature. A closer look at the plot reveals the
`glass transition temperature.
`
`Tg = 105 ºC
`
`1.51
`
`1.50
`
`1.49
`
`1.48
`
`Index of refraction, n
`
`1.47
`
`0
`
`20
`
`40
`
`60
`80
`Temperature, T
`Figure 12.4 Index of refraction as a function of temperature for PMMA (λ = 589.3 nm)
`
`100
`
`ºC
`
`140
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 533
`
`

`

`$%(
`
` 12 Optical Properties of Polymers
`
` (cid:132) 12.2 Photoelasticity and Birefringence
`
`(12.4)
`
`(12.5)
`
`Photoelasticity and flow birefringence are applications of the optical anisotropy of
`transparent media. When a transparent material is subjected to a strain field or a
`molecular orientation, the index of refraction becomes directional; the principal
`strains ε1 and ε2 are associated with principal indices of refraction n1 and n2 in a
`two-dimensional system. The difference between the two principal indices of
`refraction (birefringence) can be related to the difference of the principal strains
`using the strain-optical coefficient, k, as
`−(
`)
`n
`n
`k
`ε ε
`−
`=
`1
`2
`1
`2
`or, in terms of principal stress, as
`(
`)
`n
`n
`C
`σ σ
`−
`=
`−
`1
`2
`1
`2
`where C is the stress-optical coefficient.
`Double refractance in a material is caused when a beam of light travels through a
`transparent medium in a direction perpendicular to the plane that contains the
`principal directions of strain or refraction index, as shown schematically in Fig.
`12.5 [1]. The incoming light waves split into two waves that oscillate along the two
`principal directions. These two waves are out of phase by a distance δ defined by
`−(
`)
`n
`n t
`1
`2
`
`δ =
`
`
`
`(12.6)
`
`To observer
`
`(cid:98)= (n1-n2)t
`
`(cid:161)2
`
`n2
`
`(cid:161)1
`
`n1
`
`M
`
`t
`
`Direction of
`vibration
`
`Direction of
`propagation
`Figure 12.5 Propagation of light in a strained transparent medium
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 534
`
`

`

`
`
`!"." Photoelasticity and Birefringence
`
`$%)
`
`where t is the thickness of the transparent body. The out-of-phase distance, δ,
`between the oscillating light waves is usually referred to as the retardation.
`In photoelastic analysis, the magnitude of the stresses is determined by measuring
`the direction of the principal stresses or strains and the retardation. The technique
`and apparatus used to perform such measurements is described in the ASTM D
`4093 test. Figure 12.6 shows a schematic of such a set-up, composed of a narrow
`wavelength band light source, two polarizers, two quaterwave plates, a compen-
`sator, and a monochromatic fi lter. The polarizers and quaterwave plates must be
`perpendicular to each other (90°). The compensator is used to measure retarda-
`tion, and the monochromatic fi lter is needed when white light is not suffi cient to
`perform the photoelastic measurement. The set-up presented in Fig. 12.6 is gener-
`ally called a polariscope.
`The parameter used to quantify the strain fi eld in a specimen observed through a
`polariscope is the color. The retardation in a strained specimen is associated with
`a specifi c color. The sequence of colors and their respective retardation values and
`fringe order are shown in Table 12.1 [1]. The retardation and color can also be
`associated to a fringe order using
`Fringe order = δ
`
`λ
`
`(12.7)
`
`Monochromator
`
`Quarterwave plates
`
`Specimen
`
`Compensator
`Polarizers
`
`Light source
`
`Figure 12.6 Schematic diagram of a polariscope
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 535
`
`

`

`$%*
`
` 12 Optical Properties of Polymers
`
`Table 12.1 Retardation and Fringe Order Produced in a Polariscope
`Color
`Retardation (nm)
`Fringe order
`Black
`   0
`0
`Gray
` 160
`0.28
`White
` 260
`0.45
`Yellow
` 350
`0.60
`Orange
` 460
`0.79
`Red
` 520
`0.90
`Tint of passage
` 577
`1.00
`Blue
` 620
`1.06
`Blue-green
` 700
`1.20
`Green-yellow
` 800
`1.38
`Orange
` 940
`1.62
`Red
`1050
`1.81
`Tint of passage
`1150
`2.00
`Green
`1350
`2.33
`Green-yellow
`1450
`2.50
`Pink
`1550
`2.67
`Tint of passage
`1730
`3.00
`Green
`1800
`3.10
`Pink
`2100
`3.60
`Tint of passage
`2300
`4.00
`Green
`2400
`4.13
`
`A black body (fringe order zero) represents a strain free body, and closely spaced
`color bands represent a component with high strain gradients. The color bands
`are generally called the isochromatics. Figure 12.7 shows the isochromatic fringe
`pattern in a stressed notched bar. The fringe pattern can also be a result of mole-
`cular orientation and residual stresses in a molded transparent polymer compo-
`nent. Figure 12.8 shows the orientation induced fringe pattern in a molded part.
`The residual stress-induced birefringence is usually smaller than the orientation-
`induced pattern, making them more difficult to measure.
`Flow induced birefringence was explored by several researchers [2–4]. Likewise,
`the flow induced principal stresses can be related to the principal refraction indi-
`ces. For example, in a simple shear flow this relation can be written as [5]
`C
`C
`2
`2
`ηγ⋅
`−(
`) =
`n
`x
`x
`sin
`2
`2
`sin
`where x is the orientation of the principal axes in a simple shear flow.
`Figure 12.9 [6] shows the birefringence pattern for the flow of linear low-density
`polyethylene in a rectangular die.
`
`n
`
`2
`
`1
`
`τ
`12
`
`=
`
`(12.8)
`
`MacNeil Exhibit 2178
`Yita v. MacNeil IP, IPR2020-01139, Page 536
`
`

`

`
`
`!"." Photoelasticity and Birefringence
`
`$%+
`
`Yellow
`Red
`White
`Purple
`Grey
`Blue
`
`Lowest stress
`
`Highest stress
`
`Pink
`Pale green
`Pink
`Pink Pale green
`White
`Green
`Green
`Yellow
`
`Red
`
`Red
`
`Pale yellow
`Emerald
`Red
`Yellow
`Green
`
`Figure 12.7 Fringe pattern on a notched bar under tension
`