Fluores Monitor Injection Model Pes 97

Fluorescence Monitoring of Polymer Injection Molding: Model Development ANTHONY J. BUR Polymers Division National Institute of Standards and Technology Gaithersburg, Maryland 20899 and CHARLES L. THOMAS Department of Mechanical Engineering University of Utah Salt Lake City, Utah 84112 We have developed models to describe the behavior of an optical fiber sensor which was used to detect fluorescence from a polymer resin during the cooling phase of injection molding. The optical fiber sensor was positioned at the wall of the mold cavity by using the ejector pin channel as access to the cavity. The sources of fluorescence were dyes, which were chosen because of their sensitivity to temperature and which were mixed with the resin at dopant concentrations (parts per million by weight). The behaviors of a molecular rotor dye, dimethylamino diphenyl hexatriene, doped into polyethylene, and an excimer producing dye, bis-(pyrene) propane, doped into polystyrene were the subjects of the modeling calculations. The models consist of two modules: (a) a solution to the thermal diffusion equation for the resin cooling in the mold and (b) using temperature/time profiles and, in the case of polyethylene, crystallinity/time profiles obtained from the thermal diffusion equation, fluorescence intensity as a function of time was computed. Factors incorporated in the models are: adiabatic heating and cooling, light scattering due to microcrystals of polyethylene, crystallization kinetics, temperature and pressure shift factors for viscoelastic relaxation near the glass transition temperature of polystyrene, and the thermal resistance at the resin/mold interface.

INTRODUCTION n previously published work from this laboratory, we presented real-time measurements of fluorescence intensity during the cooling phase of injection molding of polystyrene and polyethylene (1). Fluorescence spectra were obtained using the fiber optic sensor shown in Fig. 1. Access to the mold cavity is accomplished by utilizing the ejector pin channel. The sensor consists of a bundle of optical fibers inserted into a sleeved ejector pin with a sapphire window at its end. The optical fibers view the resin through the sapphire window placed flush with the mold cavity surface. The optical fiber cable contains nineteen 100 ␮m diameter fibers, six of which transmit excitation light to the resin and thirteen of which collect the generated fluorescence light. The source of fluorescence was a fluorescent dye which was mixed with the resin at dopant levels of concentration (⬃10 parts per million by weight) prior to processing. Two tempera-

I

1430

ture sensitive fluorescent dyes, whose molecular structures are shown in Fig. 2, were used: a molecular rotor dye, dimethylamino diphenyl hexatriene (DMA DPH), and an excimer producing dye, bis-(pyrene) propane (BPP) (1–3). Real-time data, which is the subject of modeling developed in this paper, were published previously and are shown in Figs. 3 and 5 (1). The data show fluorescence intensity If versus time for DMA-DPH doped into polyethylene (Fig. 3) and the ratio of excimer to monomer intensities Iex/Im for BPP doped into polyethylene (Fig. 5). Time t ⫽ 0 corresponded to the instant of mold filling by the resin at the position of the optical sensor. Immediately thereafter, the resin began to cool and subsequent solidification of the resin was reflected in the If and Iex/Im versus time responses. In the case of the BPP doped into polystyrene, we observed that, below the glass transition temperature Tg, the production of excimer fluorescence was re-

POLYMER ENGINEERING AND SCIENCE, SEPTEMBER 1997, Vol. 37, No. 9

Fluorescence Monitoring of Polymer Injection Molding

Fig. 1. The arrangement of the optical fiber sensor in the injection mold is shown.

Fig. 3. Fluorescence intensity of DMA-DPH doped into polyethylene is plotted against time during injection molding. The excitation wavelength was 400 nm. Polymer crystallization is indicated by the short plateau at 18 seconds.

Fig. 2. (a) The fluorescent molecule, bis-(pyrene) propane (BPP), is shown. The arrows indicate rotational flexibility about the propane linkage. (b) The fluorescent dye, (1-(4-dimethylamino)-6-phenyl-1,3,5 hexatriene) (DMA-DPH) is shown. The arrow indicates rotational mobility about the endgroup.

duced to zero within the sensitivity of our measurement. (The data of Fig. 5 are not zero for T ⬍ Tg because of a background signal that has not been subtracted out.) This is seen in Fig. 5 at t ⬇ 14 s where the slope of the curve becomes zero. For DMA-DPH doped into polyethylene, we observed that fluorescence intensity increased for the first 18 s as the resin cooled, then assumed a plateau for several seconds after which the intensity resumed its monotonic increase until the resin approached ambient temperature at long times.

The short plateau at 18 to 20 s was attributed to an increase in temperature of the resin due to the heat of crystallization. For the data of Fig. 3, DMA-DPH doped into polyethylene, the increase in If with time as the resin cooled is the result of the temperature dependence of molecular rotor dyes, which is given by the DebyeEinstein equation,

If ⫽ C

␩ T

(1)

where C is a constant of proportionality, ␩ is the microviscosity in the molecular neighborhood of the dye and T is temperature. The ratio ␩/T describes not only the fluorescence intensity behavior but also the intramolecular rotational relaxation time ␶r of the dimethyl amino end group with respect to the diphenyl

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Anthony J. Bur and Charles L. Thomas hexatriene portion of the molecule. To produce fluorescence, incident excitation light energy at 400 nm raises the DMA-DPH dye to an excited state from which it decays via dissipation of energy as fluorescence radiation at 470 to 530 nm or intramolecular rotation. For short relaxation times or small ␩/T ratios (high temperature/low viscosity media), a relatively large amount of excited energy is dissipated as intramolecular rotational motion and the fluorescence intensity is diminished. Qualitatively, we see this effect in the data of Fig. 3 where the overall increase in If is due to the increase in the ratio ␩/T as the resin temperature decreases. The discontinuity in the slope of the curve of Fig. 3 at 18 s is a key event in the process because it signals the generation of the heat of crystallization as polyethylene crystallizes. The overall profile of the fluorescence/time curve is influenced by factors other than crystallization, such as large temperature gradients, thermal transport characteristics of the resin and mold, the rate at which heat of crystallization is generated, and packing pressure exerted on the resin in the mold cavity. In order to monitor packing pressure, the mold was instrumented with a pressure transducer; typical mold cavity pressure versus time data for polyethylene are shown in Fig. 4. For glass forming polystyrene, BPP was used to detect the glass transition, Tg. Its sensitivity to the onset of the glass phase is observed in the ratio of the excimer to monomer fluorescence intensity as shown in Fig. 5. The excimer fluorescence is fluorescence radiation from the excimer state, which forms when the two pyrene rings rotate into a position of close molecular contact. Upon absorption of excitation light energy at 345 nm by one pyrene ring, BPP can decay to its ground state via monomer radiation from the lone excited pyrene at 380 and 405 nm or by way of excimer radiation in the range 450 to 550 nm. The rate of excimer formation is proportional to the reciprocal of the intramolecular rotational relaxation time of the

Fig. 5. Iex/Im for bis-(pyrene) propane (BPP) doped into polystyrene are plotted against time during injection molding. Here, Iex is the excimer intensity and Im is monomer intensity from BPP. The excitation wavelength was 345 nm. Polymer solidification is indicated by the zero slope of the fluorescence vs time curve at 14 seconds. Separation of the specimen from the optical sensor corresponded to the small increase at 21 seconds.

pyrene ring motion. This rotation sweeps out a relatively large volume and requires cooperatively with the molecular dynamics of the host resin and the availability of large free volume cells (4). The cooperativity between dye and resin molecular dynamics was borne out by the observation that Iex/Im followed the Williams-Landel-Ferry (WLF) equation above Tg (1). The onset of Tg, which can be described as the collapse of free volume cell size as temperature approaches Tg, suppresses the formation of the excimer state as the BPP molecule is “frozen” into the glass matrix. For the data of the Fig. 5, the onset of Tg was attributed to the time at which the slope of the curve became zero, t ⬇ 14 s. Mold cavity pressure, which significantly influences resin molecular dynamics, was measured and is shown in Fig. 6. The development of models is the primary objective of this paper. Modeling the real-time response of these dyes involves obtaining a solution to the thermal diffusion equation for the resin cooling in the mold, and subsequently, using the temperatures and crystallinities thus obtained, to calculate the expected fluorescence intensity. Preliminary modeling results have been published (5). EXPERIMENTAL PROCEDURE

Fig. 4. Mold cavity pressure versus time is plotted for injection molding of polyethylene. 1432

The experimental procedure for characterizing DMA-DPH and BPP doped into polyethylene and polystyrene and for acquiring real-time data was described in our previous publication (1). Briefly, the resin materials were polystyrene from Fina Corp., PS 525P1 in pellet form, and high-density polyethylene from Phillips Corp., Marlex TR885 in pellet form (7). From density measurements, we determined the crystallinity of the molded polyethylene to be 0.73 (6). The fluores-

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Fluorescence Monitoring of Polymer Injection Molding heating resulting from the applied pressure. Second, the temperature and crystallinity profiles obtained from the thermal diffusion equation solution were used in the calculation of fluorescence intensity. In general, the fluorescence intensity calculation was carried out by summing the contributions over the thickness of the specimen being probed weighted by absorption, fluorescent dye temperature and pressure dependence, light scattering, and geometrical factors. Thus,

If ⫽

冕

d

␤ e ⫺␤ z 䡠 FLR ( T )

0

䡠 FLR ( P ) 䡠 SCATT 关 ␹ ( z )兴 䡠 GEOM( z ) dz Fig. 6. Mold cavity pressure versus time is plotted for injection molding of polystyrene.

cent dyes, DMA-DPH and BPP, were obtained from Molecular Probes and used as received (7). For DMADPH excitation was carried out at 400 nm and emission was detected over the wavelength range from 470 to 530 nm, and for BPP, excitation at 345 nm produced monomer emission at 380 nm and excimer emission over the range from 450 to 550 nm. The fluorescence spectra for both dyes have been published in a paper by Wang et al. (8). BPP was mixed with polystyrene by using a common solvent and dispersing doped resin with undoped resin so that the average concentration of dye was 10 ppm by weight. For polyethylene, pellets were coated with DMA-DPH and the coated pellets were mixed with uncoated pellets so that the final concentration of dye was 10 ppm by weight. The measurement system consisted of the optical fiber sensor (shown in Fig. 1), a bifurcated optical fiber cable, a xenon arc lamp light source, and a detector containing the appropriate filters and photomultiplier tubes. The bifurcated optical cable contained a bundle of fused silica fibers, a third of which are used to carry the excitation light to the resin and the other two thirds to carry the fluorescence light to the detector. The molded product was a tensile specimen 3.175 mm (1/8 inch) thick, 2 cm wide, and 10 cm in length. The view of the molded resin by the optical sensor was across the thickness. The mold was also fitted with a pressure transducer by which we monitored cavity pressure. The estimated uncertainty of the fluorescence intensity measurements was ⫾1%. The estimated uncertainty of the pressure measurements is ⫾0.02 MPa. MODEL DEVELOPMENT The model consists of two modules. First, the thermal diffusion equation was solved by the method of finite differences. For polyethylene, the thermal diffusion equation included terms for the heat of fusion generated during crystallization and for adiabatic

(2)

where If is fluorescence intensity, the first term under the integral, ␤e⫺␤z, is the probability that excitation light will be absorbed by the fluorescent dye, ␤ is the absorption coefficient of the dye, FLR(T) is the fluorescence intensity temperature dependence, FLR(P) is a pressure dependence factor, SCATT[␹(z)] is the light scattering effect caused by crystallization (␹ ⫽ crystallinity) in the resin, and GEOM(z) is a function that describes the geometrical optics of light collection and divergence. We consider the optical arrangement shown in Fig. 1. There are no lenses or focusing optics so that the light beam dimensions are determined by the exit characteristics of the light guide. The fluorescence collection efficiency is determined by the aperture of the fiber optic sensor. Two geometrical factors were considered in the model. First, as excitation light emerges from the end of the fiber, it diverges and consequently its intensity decreases with distance z of penetration into the resin. Since fluorescence intensity is directly proportional to excitation intensity, then the amount of fluorescence will decrease as a function of z. Second, we consider the optics of light collection of the fluorescence as a function of z. Each point source of fluorescence radiates fluorescence light at all 4␲ steradian of solid angle, but only a fraction of that is collected by the fiber sensor. We assume that the collection efficiency falls off as 1/z2, which is proportional to the solid angle subtended by the cross section of the fiber bundle with respect to the point source of fluorescence, i.e. the factor Ab/(zo ⫹ z)2, where zo is the thickness of the sapphire window and Ab is the cross-sectional area of the fiber bundle. Effects of light scattering and attenuation due to crystallization of polyethylene, the influence of packing pressure on crystallization kinetics and adiabatic heating, and viscoelastic volume relaxation of polystyrene as it approaches Tg are specific factors that we take into account in the model description. The applied packing pressure plays a prominent role because it affects three phenomena: the crystallization kinetics, adiabatic heating, and intramolecular rotation of the fluorescent dyes. Below, we carry out separate model calculations for polyethylene and polystyrene

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Anthony J. Bur and Charles L. Thomas because their crystalline and amorphous character require specific model development. Polyethylene. To simulate cooling in the mold, a one dimension heat flow equation was solved. That is,

␳Cp

⭸T ⫽ ⵜ 䡠 ( kⵜT ) ⫹ q˙ ⭸t

(3)

where Cp is specific heat, k is thermal conductivity and q˙ is the rate of heat generated by crystallization and by adiabatic heating and cooling, i.e., q˙ ⫽ q˙c ⫹ q˙a . q˙c can be obtained from the product of heat of fusion and the derivative of crystallinity ␹ with respect to time which is calculated from the Avrami equation,

␹ ⫽ 1 ⫺ e ⫺Kt

3

(4)

where K is the Avrami rate constant. Thus,

冋

dK dt

册

(5)

where we have used Eq 4 to eliminate t. Cp, K, and k values and their temperature dependencies for polyethylene were obtained from Van Krevelen (9). The effects of the applied packing pressure will be seen in the thermal diffusion equation via the Avrami rate constant K and adiabatic heating q˙a . K for spherulitic crystal growth is given as

K⫽

4␲ Nv 3 3

(6)

where N is the number of spherulites per cm3. v is the radial velocity of the sphere, which is expressed as

v ⫽ v o e ⫺E a/RT 䡠 e ⫺W/kT 䡠 e ⫺␥ pP

(7)

where Ea is the activation energy for diffusive transport of a polymer chain at the crystal/amorphous interface, W is the crystal surface nucleation work factor, P is pressure, and ␥p is the pressure shift factor. Values of Ea and W were taken from Van Krevelen (9) and vo and N were assigned values of 105 cm/s and 106 cm⫺3 (9). We see from Eq 7 that pressure effectively increases the activation energy for diffusive transport, but, because polyethylene is far above its Tg during the process, the effect is small. This is borne out by our calculations, which show that ␥p ⫽ 0. Equations 6 and 7 describe crystalline growth after crystal nucleation has occurred. Considerable supercooling of the melt is necessary in order to achieve nucleation. At atmospheric pressure, we observed a nucleation temperature Tn ⫽ 108°C (1). At elevated pressures, the melting temperature Tm and correspondingly Tn increase (10). In our calculation, we assumed that Tn is a linear function of pressure, Tn ⫽ 381 K ⫹ 0.6 (K/MPa)P, where P is expressed in MPa. This translates into a statement in the program code that d␹/dt ⫽ 0 when T ⬎ 381 K ⫹ 0.6 (K/MPa)P. 1434

⌬T ⫽

Tv ␣ 䡠 ⌬P Cp

(8)

where v is specific volume and ␣ is the coefficient of volume thermal expansion (11). q˙a is calculated from

q˙ a ⫽ ␳ C p T˙ ⫽ T ␣ P˙ .

(9)

The specific volume as a function of crystallinity is expressed as

v ⫽ ␹vc ⫹ (1 ⫺ ␹)va

d␹ ⫽ ( 1 ⫺ ␹ ) nK ( ⫺K ⫺1 ln( 1 ⫺ ␹ )) (n⫺1)/n dt ⫹ ( ⫺K ⫺1 ln( 1 ⫺ ␹ ))

q˙a is created in the process by the application and release of packing pressure. In an adiabatic process, the temperature change ⌬T associated with a change of applied pressure ⌬P is given by

(10)

where va and vc are amorphous and crystalline specific volumes. ␣ ⫽ (1/v)(dv/dT ) is obtained from Eq 10 using data of Davis et al. for the value of dvc/dT (12). For va, we use the Hartmann-Haque equation of state (13),

( P/B o )( v a /V o ) 5 ⫽ ( T/T o ) 3/ 2 ⫺ ln( v a /V o )

(11)

where Bo ⫽ 2800 MPa, Vo ⫽ 1.0362 cm3/g, and To ⫽ 1203 K for linear polyethylene. Other potential effects caused by applied pressure such as effects on heat of fusion, on the viscosity term of Eq 1, and on Cp and k were neglected. Measurements of Karasz and Jones show that heat of fusion is a weak function of pressure (14). The viscosity of Eq 1 is not affected because the intramolecular motion of the DMA-DPH dye is viewed as a local or ␤ relaxation for which pressure effects can be neglected (1, 15). Designating the DMA-DPH motion as a ␤ relaxation originates from our observation that, upon cooling polystyrene doped with DMA-DPH through its glass transition, no indication of the onset of the glass phase is seen in the fluorescence intensity (1). We conclude that the motion of the dimethylamino end group does not correlate with the macromolecular chain dynamics (the ␣ relaxation) at temperatures ⬎Tg. The initial conditions were a resin temperature of 200°C and a steel mold temperature of 25°C. The assumed boundary conditions, depicted in Fig. 7, were that the extreme outer edge of the mold (steel/air interface) is an insulating boundary and that, because of symmetry, the centerline of the resin at 0.1587 cm (1/16 inch) is also an insulating boundary. Thus, a solution can be obtained from only half of the problem as shown in Fig. 7(b). Another thermal boundary to be defined is the interface between resin and mold. Here we assume that the heat flux is continuous across this interface and that there exists a thermal resistance which impedes the transfer of heat from resin to mold. We describe the thermal resistance in terms of a thermal transfer

POLYMER ENGINEERING AND SCIENCE, SEPTEMBER 1997, Vol. 37, No. 9

Fluorescence Monitoring of Polymer Injection Molding

Fig. 7. The boundary conditions for solving the thermal diffusion equation are displayed. Symmetry of the problem permits an insulation boundary to be placed at the center line of the resin.

coefficient h where,

h⫽

Q ⌬T

(12)

Here, Q is the heat flux and ⌬T is the temperature difference between resin and mold at the interface. Kamal and co-workers (16) found that h was a function of time and could be expressed as an exponential decay. For our calculations, we used

h ⫽ h o e ⫺t/ ␶ ⫹ h ⬁

(13)

where ho ⫽ 5h⬁ and ␶ ⫽ 0.4 s. Below, we describe how a value of h⬁ was deduced from our calculations by using it as a parameter for fitting the calculated results to experimental data. With the boundary and initial conditions cited above, a one-dimensional thermal diffusion equation was solved by the method of finite differences for both the steel mold and the resin. The one dimensional path z was divided into 81 elements where elements 0 and 80 were at the resin/mold interfaces. Fluorescence intensity was calculated by summing the contributions from the 81 elements modified by the appropriate geometrical and attenuation factors. As indicated in Fig. 8 there are two paths over which light can traverse: excitation light can propagate directly to the i th layer where it is absorbed or it can reflect off the back surface before being absorbed at the i th layer; likewise, the fluorescence from the i th layer can propagate directly to the optical fiber sensor or it can reflect from the back wall of the mold before being collected. Light attenuation due to scattering by microcrystals in high-density polyethylene will be taken into account by considering two effects: (a) Scattering of light produced by growing spherulites, which scatter light

because of the difference in index of refraction between spherulites and the surrounding amorphous material, and (b) scattering caused by microcrystals within the spherulites. Scattering due to growing spherulites in polyethylene was studied by Stein and co-workers, who observed that scattering increases as the spherulite grows, reaches a maximum, and returns to zero at the completion of crystallization (17, 18). Scattering becomes small at the end of crystallization because, at this stage, the spherulites are entirely volume filling, i.e., there is little amorphous material to offer a differential index of refraction. On the other hand, scattering by the microcrystals within the spherulites increases with crystallinity, and remains finite at the end of crystallization because the microcrystals are separated by interfacing amorphous material. The effect of light scattering on the transmitted light through the injection molded product at 73% crystallinity was measured. The data of Fig. 9 are from 1-mm-thick specimens that were placed sequentially in series while the light transmission was measured. Here, the reciprocal of transmitted light versus thickness is plotted. The linear behavior demonstrates that light transmission is diffusion-like in high-density polyethylene and results from multiple scattering and reflection of photons. In the diffusion regime, light transmission is described by the function 1/z, which is a solution to the steady-state Laplace equation. In the model, the transmission function is expressed as 1/[␹(z)z], i.e., we consider the product ␹(z)z to be the characteristic scattering length where ␹(z) is the crystallinity at z. Consideration of the above factors yields the following expression for fluorescence intensity from highdensity polyethylene of thickness d containing a mo-

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Anthony J. Bur and Charles L. Thomas

Fig. 9. The reciprocal of light transmitted through high-density polyethylene is plotted versus path length where Io is incident light and I is transmitted light. The estimated uncertainty of Io/I is ⫾0.05.

scattering plus attenuation due to light diffusion; ␣ is the Stein scattering function to be discussed below and ␹ is crystallinity; the fourth term expresses the effect of excitation light divergence where Ao is the cross-sectional area of the excitation beam at z ⫽ 0 and A(z) is the cross-sectional area of the excitation beam at position z; the fifth term depicts the collection efficiency of a fiber bundle with cross-sectional area Ab for which light collection falls off as 1/z 2; and zo is the thickness of the sapphire optical window. The Stein scattering function ␣ is given as

I s ⫽ ␣ I o ⫽ ( D ␾ s ⫺ E ␾ 2s ) I o

(15)

where Is is scattered light, Io is the incident light, D and E are constants that depend on polarizability and spatial orientation of the microcrystal, and polarizability of the surrounding amorphous material. ␾S is the volume fraction of the spherulite (17, 18). Attenuation of the transmitted light It can be expressed as Fig. 8. Light paths including reflection from the back surface of the mold for (a) the excitation light and (b) fluorescence light are shown.

冕

d

0

␤ e ⫺␤ z 䡠

冋

␩ (1 ⫺ C) 䡠 Ce ⫺␣ z ⫹ T ␹z 䡠

册

Ao Ab 䡠 dz A( z) ( zo ⫹ z)2

(14)

Here, B and C are constants. The first term under the integral is the probability that excitation light is absorbed; the second term, ␩/T, is the molecular rotor dye modulation term as noted in Eq 1; the third term is the light scattering attenuation factor, which describes attenuation as the sum of Stein spherulite 1436

dI t ⫽ 关 D ␾ s ⫺ E ␾ 2s 兴 I t dx

(16)

We note that ␾S is not equal to crystallinity ␹ because polymer crystal spherulites are known to be ⬍100% crystalline. For these calculations, spherulite crystallinity was set equal to the specimen crystallinity, 73%. The temperature dependence of viscosity ␩ is expressed as

lecular rotor dye,

If ⫽ B

⫺

␩ ⫽ ␩ o e ⌬H/RT

(17)

where ⌬H is the activation energy for the intramolecular motion of the dye molecule and R is the universal gas constant. Our measurement of ⌬H for DMA-DPH in polyethylene was 2.09 ⫻ 104 J/mole (5 kcal/mole). The solution of the thermal diffusion equation yielded both temperature/time/position and crystallinity/time/position profiles. These results are shown

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Fluorescence Monitoring of Polymer Injection Molding in Figs. 10 and 11. In the case of the calculated temperature/time profile, Fig. 10, the data are shown for five equi-spaced positions in the resin from the skin (the resin/mold interface) to the inner core. The effects of adiabatic heating and the heat of crystallization are readily seen. The skin temperature falls rapidly to the ambient temperature after a small increase due to heat of crystallization and the decrease in the value of h. The core temperature initially increases (adiabatic heating) and then decreases to a plateau value of ⬃127°C for 10 s. This is because the heat of crystallinity, generated by the crystallizing resin at positions closer to the skin, impedes the transport of heat from the core. Upon crystallization of the core at 20 s, the core temperature drops relatively quickly to ambient temperature. The growth of the crystallinity is seen in Fig. 11, where it is demonstrated that the crystallization front proceeds from the skin to the core in ⬃20 s. We note that the calculations were carried out for a thermal transfer coefficient at the resin/mold interface of h ⫽ 0.25e⫺t/0.4 ⫹ 0.05 (J/cm2s°C). The effect of varying h⬁ is shown in Fig. 12, where we have plotted average crystallinity versus time for a range of h⬁ values. The lower the coefficient, the longer it takes to reach the final crystallization at 73%. It is clear that h has a marked effect on the elapsed time for crystallization. Using the crystallinity and temperature calculations of Figs. 10, 11, and 12, the fluorescence intensity was calculated from the following expression:

冘 80

If ⫽ B

i⫽0

Fig. 11. Calculated crystallinity versus distance from skin to the central core is shown at a series of times from 2 to 18 s. The calculations were carried out for the thermal transport coefficient h⬁ ⫽ 0.05 J/cm2s°C.

e ⌬H/RT ri 䡠 ␤ ⌬ze ⫺␤ i⌬z T ri

冋

䡠 Ce ⫺兺 j⫽02 ␣ j⌬z ⫹

册

1⫺C 䡠 ( GEOM) i N兺 k⫽0 ␹ k ⌬z

(18)

Fig. 12. The calculated average crystallinity of high-density polyethylene versus time is shown for a range of thermal transport coefficient h⬁ values from 0.015 to 10 J/cm2s°C.

Fig. 10. Calculated temperature versus time profiles for highdensity polyethylene are shown. The curves are for the skin (resin and mold interface), the central core and for 25, 50, 75% of the thickness between them. The calculations were made for the thermal transport coefficient h⬁ ⫽ 0.05 J/cm2s°C.

where B and C are amplitude factors, and (GEOM) is the geometrical correction factor for light divergence and fiber collection efficiency as described above. N is a normalization constant that is adjusted so that there is no scattering when ␹ ⫽ 0. (Of course, ␹ is not permitted to be zero for this calculation and therefore, we establish a minimum crystallinity of 0.001.) For simplicity of presentation, reflection off the back surface of the mold is not included in the Eq 18, but was included in our calculations; it involves the addition of terms similar to Eq 18. The calculated fluorescence intensity due to reflections was ⬃10% of the total fluorescence intensity at t ⫽ 0 dropping to 5% at the end of crystallization. For the calculation of If , we used the following constants: ␩o ⫽ 1 Pa 䡠 s, ␤ ⫽ 1 cm⫺1, ⌬H ⫽ 2.093 ⫻ 104 joules/mole (5 kcal/mole), Ao ⫽ 2.516 ⫻

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Anthony J. Bur and Charles L. Thomas 10⫺2 cm2, Ab ⫽ 7.85 ⫻ 10⫺3 cm2, zo ⫽ 0.317 cm and resin thickness ⫽ 0.3175 cm (1/8 inch). The density was obtained from the reciprocal of Eq 10. Values of constants D and E of Eq 15 were estimated to be 20 cm⫺1 from light transmission measurements made on low-density polyethylene. Polypropylene, instead of polyethylene, was used to obtain these numbers because light transmission measurements on polypropylene showed a distinct minimum, which was assumed to occur at ␾ ⫽ 0.5 (19). High-density polyethylene, on the other hand, does not display the minimum because of its highly diffusive transmission character. In any case, the calculations are somewhat insensitive to the values of D and E; changing their values by 100% yield results that differ by ⬍0.5%. Calculated results of fluorescence intensity If versus time for a range of h⬁ values are shown in Fig. 13(top). The effect on If caused by the heat of crystallization, the release of pressure and crystallization kinetics is

Fig. 13. (Top) Calculated fluorescence intensity is plotted versus time for a range of thermal transport coefficient h values from 0.015 to 10 J/cm2s°C; (bottom) The calculation of Figure 11 is shown in order to compare with the fluorescence intensity calculations. The knee in the calculated fluorescence curves corresponds to the end of crystallinity. 1438

seen in the knee of the curve. When the If calculations are compared with the crystallinity data of Fig. 13 (bottom), we see that the knee in the If curves is coincident in time with the completion of crystallinity. Using h⬁ as an adjustable parameter, the calculated If was made coincident in time with the observed data (Fig. 3) by shifting the knee in the calculated curve to 20 s. A value of h⬁ ⫽ 0.05 J/cm2s°C was deduced from this curve fitting procedure. The calculated fluorescence intensity was adjusted using the amplitude constants B and C. At t ⫽ 0, we assume that ␹ ⫽ 0 and therefore the scattering term in brackets in Eq 18 equals 1. For t ⫽ 0, B is adjusted so that Icalc(0) ⫽ Imeas(0). Using this value of B, C is adjusted so that the calculated change in If is equal to the measured change, i.e., ⌬Icalc ⫽ ⌬Imeas. These adjustments in B and C force the calculated and measured intensities into equality at the beginning and end of the time profile. The quality of fit is then judged by comparing the shapes and intensities of the calculated and measured curves in the vicinity of the knee in the curve. The result is shown in Fig. 14. The calculated and observed curves have identical shape but differ in magnitude by ⬃12% at the knee. The concave upward shape of the curves for t ⬍ 18 s is caused by a combination of adiabatic cooling, heat of crystallization, and temperature dependent crystallization kinetics. The pressure shift factor ␥p for this calculation was zero. In order to account for the small differences between calculated and observed results of Fig. 14, several adjustments to the model were considered: (a) creating a crystallinity gradient in the specimen whereby the skin has a crystallinity 20% lower than the core, (b) incomplete spherulitic growth for which ␾s ⬍ 1 at the completion of crystallization, (c) a pressure shift factor associated with the intramolecular rotational relaxation of the dye, and (d) thermal deg-

Fig. 14. The fit of the calculated fluorescence to the real-time observed data is shown. The best fit was obtained for h⬁ ⫽ 0.05 J/cm2s°C.

POLYMER ENGINEERING AND SCIENCE, SEPTEMBER 1997, Vol. 37, No. 9

Fluorescence Monitoring of Polymer Injection Molding radation of the dye at high temperatures, which caused the concentration of dye at the core to decrease with time. We found that reasonable magnitudes of these adjustments yielded results that differed by no more than 5% from the calculation of Fig. 14 and were qualitatively of similar shape. The calculations are more sensitive to the functional form of h, the thermal transfer coefficient, but we have not studied this extensively. We have used an exponential function, Eq 13, for h as measured by Kamal (16). There is a need for a more thorough experimental investigation of h and its dependence on resin, mold surfaces, and operating conditions. The effect of scattering on fluorescence intensity is shown in Fig. 15. The curve designated “no scattering” was calculated by setting the scattering term [term in brackets of Eq 18] ⫽ 1 for all time. We see that fluorescence intensity without scattering is approximately three times larger than the observed intensity. Thus, in spite of multiple scattering by resin microcrystals, 1⁄3 of the light is transmitted. This is characteristic of transport by diffusion for which a significant amount of light is scattered in the forward direction. Polystyrene. The same general procedure used for polyethylene calculations was used for modeling realtime fluorescence observations of excimer forming BPP in polystyrene. That is, we solved the heat diffusion equation in one dimension by the method of finite differences, and using the temperature arrays thus obtained, the ratio of excimer to monomer fluorescence Iex/Im was calculated. Important factors incorporated in the model were volume viscoelastic relaxation and adiabatic heating. The effect of adiabatic heating was calculated using Eq 7 and 9 and the Hartmann-Haque equation of state for polystyrene with vo ⫽ 0.8732, Bo ⫽ 3110 MPa, and To ⫽ 1581 K (13). Adiabatic cooling effects were neglected because the rate of pressure release, as seen in Fig. 6, was relatively slow.

As mentioned above, Iex/Im obeys the WLF equation at temperatures immediately ⬎Tg (1). The onset of Tg can be seen in Fig. 16, which is a plot of Iex/Im versus temperature. The knee in the curve that occurs at 105°C is our operational definition of Tg. Below Tg, the ratio Iex/Im was found to be insensitive to temperature because the glass state inhibits the formation of the BPP excimer state. Other experiments showed that, for T ⬍ Tg, Iex was zero within the limit of our detector sensitivity. Thus, it appears that the data of Fig. 16 includes background light, which must be accounted for in the model calculation. The data of Fig. 16, fitted with a polynomial function, were used as a calibration curve. For the calculation of Iex/Im, we assumed that

Fig. 15. The effect of light scattering by polyethylene microcrystals is shown in these plots of fluorescence intensity versus time for calculations with and without scattering.

Fig. 16. Iex/Im for BPP doped into polystyrene is plotted versus temperature for measurements made under equilibrium conditions.

( I ex /I m ) calc ⫽

q ri(Ti) 兺 i⫽0 1⫹1

(19)

where q ⫹ 1 is the number of finite elements used for the calculation and ri(Ti ) is Iex/Im at the i th element whose temperature is Ti. Thus, (Iex/Im)calc is taken as the average of ri. The value of each ri was obtained from the calibration curve of Fig. 16 for which temperature is the lone independent parameter. The results are shown in Fig. 17, where we have plotted the measured and calculated values of Iex/Im versus time. We note that the calculated curve descends rapidly to the plateau value, indicating that the molded polystyrene was cooled in a near quench-like process. The observed value of Iex/Im, however, approached the longtime plateau much more slowly. Another view of the rapid decrease in temperature can be seen in Fig. 18, where we have plotted calculated temperature versus time. Based on these calculated temperatures, we would conclude erroneously that the resin was in the glass phase at 7.6 sec. The difference between these calculations and our experimental observations is due to viscoelastic volume relaxation, which controlled volume contraction of polystyrene as it cooled. While the temperature descended rapidly, volume changes were controlled by the slower molecular dynamics of

POLYMER ENGINEERING AND SCIENCE, SEPTEMBER 1997, Vol. 37, No. 9

1439

Anthony J. Bur and Charles L. Thomas express ␶ of a glass forming polymer as a function of temperature, pressure and structure shift factors, a T , aP, and a␦ (20 –22):

␶ ⫽ ␶oaTaPa␦

(21)

Here, a␦ describes the shift in relaxation times for relaxation to equilibrium from a non-equilibrium state, where the non-equilibrium state has been created by a temperature and/or pressure jump. The shift factors are expressed as

Fig. 17. Iex/Im calculated for no relaxation effects is plotted versus time and is compared with real-time Iex/Im data of Figure 5 for BPP doped into polystyrene. The jump at t ⫽ 21 s (Figure 5) has been subtracted out.

a T ⫽ e ⫺␪ T ( T⫺T ref )

(22)

a P ⫽ e ␪ PP

(23)

a ␦ ⫽ e [(1⫺x) ␪ P␦ /⌬ ␣ ]

(24)

and

where Tref is a reference temperature, ␪T is a constant associated with the activation energy of the process, ␪P is the pressure coefficient, ⌬␣ is the change in the thermal expansion coefficient at the glass transition, x is a material constant ( x ⫽ 0.2 for polystyrene), and ␦ is the normalized deviation from equilibrium defined by volumes V(t) and V(⬁), i.e.,

␦(t) ⫽

V(t) ⫺ V(⬁) V(⬁)

(25)

␦(t) can be calculated from

␦(t) ⫽

冕

t

( ⫺⌬ ␣ T˙ ⫹ ⌬kP˙ ) R ( t ) dt

(26)

0

Fig. 18. The calculated temperatures at the skin and the core of polystyrene are plotted versus time. It is seen that the core temperature drops below 105°C at t ⫽ 7.6 s.

where T˙ and P˙ are the change in pressure and temperature with time. R(t) is a relaxation function defined as t

R ( t ) ⫽ e ⫺关 兰 t⬘ dt/ ␶ 兴 the polymer in the vicinity of Tg. Considering that Iex/Im obeys the WLF equation, it is not surprising that our observations show the effects of viscoelastic volume relaxation. To include relaxation effects in the model, we consider that production of excimer fluorescence is controlled by the intramolecular rotational relaxation time ␶ of the BPP molecule. Excimer fluorescence and the ratio Iex/Im are proportional to the rate constant k of excimer formation or to the reciprocal of ␶ (3). The calculation of Iex/Im was carried out by averaging over the q ⫹ 1 elements for which ki and ␶i are the rate constant and relaxation time of the i th element. We have

冉 冊 I ex Im

⫽ calc

q F 兺 i⫽0 ( 1/ ␶ i ) q⫹1

(20)

where F is a constant of proportionality. The manner in which ki and ␶i depend on temperature, pressure, and volume at the i th element can be described by the model of Kovacs and co-workers, who 1440

␤

(27)

where ␤ is a constant of the Williams-Watts stretch exponential (20, 21). From Eqs 24 through 27, we note that a␦ is a function of ␶ so that the calculation of ␶ for use in Eq 20 involved an iteration technique. For the calculation, the following constants were used: ⌬␣ ⫽ 0.00034 K⫺1, ␤ ⫽ 0.5, ␶o ⫽ 0.01 s, Tref ⫽ 220°C, ⌬k ⫽ 2E-4 ⫺ 6.9E-8P MPa⫺1 (21). The pressure P was obtained from the data of Fig. 6. In carrying out the fit to the data, the constant F of Eq 20 was adjusted so that calculated and measured values matched at t ⫽ 0; also, a constant was added to account for the background. The resultant calculation is shown in Fig. 19. ␪T and ␪P, which were used as adjustable fitting parameters, were found to have values 0.018 K⫺1 and 0.022 MPa⫺1 for the short-dashed curve. Because ␪T and ␪P complement each other, increasing ␪T and decreasing ␪P by 20% did not affect the quality of the fit. These values of ␪T and ␪P are lower than those obtained by Tribone et al. (22), but their data was confined to T ⬍ 150°C whereas our data extends to 220°C. The long-dashed curve was calcu-

POLYMER ENGINEERING AND SCIENCE, SEPTEMBER 1997, Vol. 37, No. 9

Fluorescence Monitoring of Polymer Injection Molding

Fig. 19. Iex/Im calculated assuming relaxation effects is plotted versus time and compared with real-time Iex/Im data of Figure 5 for BPP doped into polystyrene. The jump at t ⫽ 21 s (Figure 5) has been subtracted out. Short-dashed curve: calculation for optimum fit; long-dashed curve: calculation assuming that pressure is atmospheric. The calculations were carried out for h ⫽ 0.2e⫺t/0.3 ⫹ 0.04 J/cm2s°C.

lated for P˙ ⫽ 0, i.e., pressure remains atmospheric throughout the process; the curve shows that the effects due to applied pressure are volume compression and a decrease in the rate constant for formation of the excimer state. The model calculation shows that the initial rapid decrease in excimer intensity was due primarily to the nearly stepwise buildup of pressure at t ⫽ 0.25 s (see Fig. 6); the decrease in temperature played a secondary role here. In this case, the calculation is not nearly as sensitive to the value of h as is the calculation for polyethylene. This is because h is a thermal parameter whereas BPP excimer fluorescence is primarily dependent on the volume state of the resin and the availability of free volume cells large enough to accommodate BPP photochromic activity. Pressure rather than temperature has the greater influence on the free volume of the resin, and this is reflected in the calculation. We found that h⬁ values in the range 0.04 to 0.15 J/cm2s°C yielded satisfactory agreement with data. CONCLUSIONS We have developed models for the behavior of a fluorescence optical fiber sensor used to monitor injection molding. The models consist of solving the thermal diffusion equation and using the resultant temperature and crystallinity profiles to calculate the fluorescence intensity changes. Fluorescence monitoring of polyethylene and polystyrene during the cooling phase of injection molding was modeled. The fluorescence intensity calculation takes into account the temperature sensitivity of the dye, heat of crystallization, adiabatic heating and cooling, crystallization kinetics of polyethylene, light scattering due to growing crystallites in polyethylene, viscoelastic relaxation effects in glass forming polystyrene, and geometrical correction factors. From the thermal diffusion and

fluorescence calculations we were able to deduce a value for the thermal transport coefficient at the resin/mold interface. For polyethylene, the model calculation yielded the time at which crystallization was complete and showed the approach of the resin temperature to the ambient temperature of the mold. For polystyrene, the model revealed the effect of viscoelastic volume relaxation at the onset of Tg and showed that the rapid initial decrease in excimer fluorescence was due to the stepwise application of packing pressure. We found that the thermal transfer coefficient h plays a prominent role for determining the time of solidification. Because its value has large implications for commercial processors, a careful experimental investigation of this coefficient for various resins, interfaces, and mold combinations is warranted.

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