[
    {
        "id": 1,
        "chunk": "# Kinetic studies of polyurethane polymerization with Raman spectroscopy  \n\nShane Parnell, K. Min\\*, M. Cakmak  \n\nDepartment of Polymer Engineering, University of Akron, Akron, OH 44325-0301, USA  \n\nReceived 18 September 2002; received in revised form 4 April 2003; accepted 22 May 2003",
        "category": " Introduction"
    },
    {
        "id": 2,
        "chunk": "# Abstract  \n\nIn this study, the polymerization kinetics of an uncatalyzed polyester based thermoplastic polyurethane formulation was characterized with Raman spectroscopy. Measuring the normalized scattering intensity of a band originating from the TPU diisocyanate, conversion was calculated as a function of time. Kinetic parameters obtained from these experiments correlated well with those obtained from analogous calorimetric experiments and with literature values. It was concluded that Raman spectroscopy is a powerful tool for characterizing the polymerization kinetics of polyurethanes in situ.  \n\n$\\circledcirc$ 2003 Elsevier Ltd. All rights reserved.  \n\nKeywords: Raman spectroscopy; Polyurethane; Kinetics",
        "category": " Abstract"
    },
    {
        "id": 3,
        "chunk": "# 1. Introduction  \n\nA large number of characterization methods have been used to monitor the kinetics of polymerization reactions. Kamal [1] and Mussati [2] have given extensive reviews. These methods fall into two groups: indirect methods, which measure a physical property that can be functionally related to the extent of reaction, and direct methods, which measure the concentration of reactant or product species. Rheometry and thermal methods fall into the first group while titration and spectroscopy belong to the second.  \n\nOf the indirect thermal methods used to monitor the polymerization kinetics of polyurethanes, differential scanning calorimetry [3,4] (DSC) and adiabatic temperature rise [5–9] (ATR) have the advantage that they are simple. However, given the fact that urethane systems are mixing activated, DSC can only follow slow polyurethane reactions. ATR on the other hand, can follow fast polyurethane reactions. Nonetheless, ATR is still an indirect method and many assumptions have to be made to relate heat evolution to extent of reaction.  \n\nOf the direct methods used to monitor the polymerization kinetics of polyurethanes, spectroscopic techniques have the advantage that they can measure extent of reaction directly, are capable of monitoring fast reactions, and can monitor several chemical changes at once. However, the versatility of infrared (IR) spectroscopy is limited due to sample preparation requirements [6]. Even with the use of attenuated total reflectance techniques, IR spectroscopy is still limited in that special sample cells must be constructed [10]. In contrast, Raman spectroscopy has several advantages. These advantages include minimal required sampling volume, the ability to utilize glass and other closed containers for sample cells, and larger frequency ranges for spectral observation on one instrument. However, the first and foremost advantage in Raman spectroscopy is sample preparation. Since the Raman effect is a scattering process, samples of any shape or size can be examined. Moreover, Raman spectroscopy measurements can be conducted remotely using inexpensive, communications grade, fused-silica optical fibers. A theoretical background and mathematical treatment of Raman scattering have been developed by Grasselli [11] and Koenig [12].  \n\nThese advantageous characteristics make Raman spectroscopy particularly useful for the in situ characterization of polymerization reactions where the removal of samples for off-line characterization is not always possible or practical. Since thermoplastic polyurethanes (TPUs) are produced continuously via reactive extrusion, the value of a versatile on-line characterization technique such as Raman spectroscopy becomes evident. The objective of this study is to characterize the kinetics of TPU polymerization in situ with Raman spectroscopy. More specifically, Raman spectroscopy will be used to acquire conversion versus time data from the polymerization of an uncatalyzed polyester based TPU formulation. Kinetics parameters extracted from such data will be compared to those obtained from analogous DSC experiments and literature values.",
        "category": " Introduction"
    },
    {
        "id": 4,
        "chunk": "# 2. Experimental",
        "category": " Materials and methods"
    },
    {
        "id": 5,
        "chunk": "# 2.1. Materials  \n\nThe soft segment of the TPU used throughout this study was a hydroxyl terminated poly(butylene adipate) (PBA) oligomer supplied by Bayer. This diol had a number average molecular weight of approximately $2000\\mathrm{g/mol}$ . The hard segments of the TPU were derived from $^{4,4^{\\prime}}$ -diphenylmethane diisocyanate (MDI) and 1,4-butanediol (BDO). MDI was supplied by Bayer while BDO was supplied by ARCO chemical company.",
        "category": " Materials and methods"
    },
    {
        "id": 6,
        "chunk": "# 2.2. Sample preparation  \n\nIn preparation for TPU synthesis, PBA was melted and dried under a vacuum at $100^{\\circ}\\mathrm{C}$ for a minimum of $4\\mathrm{h}$ while BDO was dried over type 3A molecular sieves at room temperature for at least 2 weeks prior to synthesis. MDI was used as received but was stored under a vacuum at $0^{\\circ}\\mathrm{C}$ until required for synthesis.  \n\nThe TPU was synthesized with a relatively low hard segment content, corresponding to equimolar quantities of PBA and BDO. Keeping the stoichiometric ratio of hydroxyl to isocyanate functionality at unity, the TPU contained $76.84\\%$ PBA, $3.536\\%$ BDO, and $19.62\\%$ MDI by mass (based on PBA with an equivalent number average molecular weight of $979.1\\mathrm{g/mol})$ . Regardless of the polymerization environment, the ‘one-shot’ process was always the preferred route of TPU synthesis.  \n\nIn this procedure, dewatered PBA (heated to $100^{\\circ}\\mathrm{C})$ , BDO (at room temperature), and MDI (at room temperature) were gravimetrically metered into a $500\\mathrm{ml}$ polypropylene beaker and vigorously hand mixed for $15\\mathrm{~s~}$ Having thoroughly mixed all TPU reactants, samples were removed and prepared for immediate kinetic analysis.",
        "category": " Materials and methods"
    },
    {
        "id": 7,
        "chunk": "# 2.3. Raman spectroscopy measurements  \n\nA Kaiser Optical Systems Series 5000 Holoprobe Raman spectrometer was used to measure the kinetics of TPU polymerization. Equipped with a thermoelectrically cooled charge coupled device (CCD) detector, the system was capable of collecting spectra over a Raman shift spectral range of approximately $300{-}3300\\mathrm{cm}^{-1}$ . A $100~\\mathrm{{mW}}$ $785\\mathrm{nm}$ GaAlAs diode laser was used as the excitation radiation source.  \n\nUsing a $180^{\\circ}$ backscattering Raman measurement geometry, TPU reactant mixture samples approximately  \n\n$1.0\\mathrm{mm}$ in thickness were sandwiched between fused quartz cover slips and placed in a hot stage preheated to a specific isothermal polymerization temperature. Upon aligning the aperture of the hot stage, and thus the sample, with the focused Raman laser beam, a 30-s exposure time was used to generate a spectrum every $30~\\mathrm{s}$ . These isothermal experiments were terminated after $60\\mathrm{min}$ . Isothermal polymerization temperatures of 100, 120, 140, and $160^{\\circ}\\mathrm{C}$ were used to evaluate all kinetic parameters.  \n\nIn preparation for quantitative analysis, all Raman spectra were processed with several chemometric spectral manipulation techniques using Grams/386 software from Galactic Industries Corp. In order to remove Raleigh/fluorescence induced background scattering, a best-fit, fourth order, polynomial baseline was subtracted from all spectra. Because Raman spectroscopy is a single beam method and because the number of scattering sites can never be known in the analysis of solids, all Raman spectra were normalized with respect to an internal standard. To this end, peak intensity of the $1612\\mathrm{cm}^{-1}$ band was used. This band was the result of aromatic ring breathing/stretching vibrational modes present in the phenylene groups of MDI.  \n\nPeak height was used as a measure of peak intensity in this study. Although peak areas are most desirable, measurements can only be made in this way when the signal-to-noise (S/N) is very high, and the baseline is well defined. Small errors have a disproportionate effect on the final result using this method because all points in the spectral peak are given equal weight in the calculation. In contrast to peak area measurements, peak height measurements usually give the best results unless there is a significant change in peak shape with concentration. If such measurements are made at a peak’s maximum, the point of optimum S/N is used reducing errors attributed to random noise. However, since this type of measurement is sensitive to high frequency noise, all Raman signals were filtered accordingly.",
        "category": " Materials and methods"
    },
    {
        "id": 8,
        "chunk": "# 2.4. Calorimetric measurements  \n\nA Thermal Advantage 2920 modulated differential scanning calorimeter operating in the isothermal mode was also used to measure the kinetics of TPU polymerization. In conducting these experiments, TPU reactant mixture samples were carefully weighed to $10\\pm2\\mathrm{mg}$ and sealed in aluminum hermetic pans and lids. Upon placing a sealed sample into the DSC preheated to a specific isothermal temperature, heat flow resulting from the exothermic TPU polymerization reaction was measured as a function of time. After $60\\mathrm{min}$ of isothermal polymerization, samples were immediately quenched to $0^{\\circ}\\mathrm{C}$ at a cooling rate of $-100\\mathrm{^{\\circ}C/m i n}$ and then subjected to a temperature scan from 0 to $200^{\\circ}\\mathrm{C}$ at a heating rate of $20\\ \\mathrm{{^circC/min}}$ . This temperature scan was performed in an effort to quantify any residual heat of reaction not evolved in the previous isothermal scan and to ensure complete TPU polymerization. Isothermal polymerization temperatures of 100, 120, 140, and $160^{\\circ}\\mathrm{C}$ were used to evaluate all kinetic parameters.",
        "category": " Materials and methods"
    },
    {
        "id": 9,
        "chunk": "# 3. Results and discussion",
        "category": " Results and discussion"
    },
    {
        "id": 10,
        "chunk": "# 3.1. TPU conversion from Raman spectroscopy  \n\nQuantitative kinetic analysis of a reacting system with Raman spectroscopy is based on measuring changes in peak intensity of bands belonging to characteristic reactant or product functional groups during the reaction period. Therefore, to elucidate which bands may be suitable for kinetic measurements on the TPU formulation investigated in this study, Fig. 1 shows partial Raman spectra of an uncatalyzed TPU reactant mixture after 1 min, $30\\mathrm{min}$ , and $12\\mathrm{h}$ $\\langle\\alpha\\cong1\\rangle$ of polymerization at $120^{\\circ}\\mathrm{C}$ . Tentative band assignments are made from reference to earlier Raman studies of polyesters [13–16], isocyanates [15,16], and urethanes [15,16] and are listed in Table 1. In principal, Raman scattering intensities of isocyanate (asymmetric stretch at $2275~\\mathrm{{cm}^{-1}}$ and symmetric stretch at $14\\dot{4}5~\\mathrm{cm}^{-1};$ ), hydroxyl, and urethane $(\\mathrm{N-H}$ stretch, amide I at ca. $1\\dot{7}32\\mathrm{cm}^{-1}$ , amide $\\mathrm{II}$ at ca. $1530\\mathrm{cm}^{-1}$ , and amide III at ca. $1303\\mathrm{cm}^{-1},$ ) functional groups can all be used to determine the kinetics of polymerization for this particular TPU formulation. However, bands resulting from hydroxyl and urethane $\\mathrm{\\DeltaN-H}$ stretching vibrations cannot be used for quantitative kinetic analysis since they are too small or fall outside of the Raman shift spectral range (ca. $300-$ $3300\\mathrm{cm}^{-1}.$ ) accessible in these experiments. Bands resulting from amide I vibrations in urethane linkages produced during TPU polymerization could be used for quantitative analysis, but these bands overlap those from carbonyl stretching vibrations present in the ester groups of PBA. This coupled with complications arising from H-bonding and low S/N ratios render quantitative measurements on bands from urethane amide I vibrations very difficult. Bands originating from urethane amide II and amide III vibrations are viable candidates for quantitative measurements, but they also suffer from multi-peak overlap and/or low S/N ratios.  \n\n![](images/4b47818368219717a2977c3c5aeba2f5cd1e5531ea98e4892e2034c738d0a0c3.jpg)  \nFig. 1. Partial Raman spectra of the TPU reactant mixture polymerized at $120^{\\circ}\\mathrm{C}$ for $1\\mathrm{min}$ , $30\\mathrm{min}$ , and $12\\mathrm{h}$ .  \n\nTable 1 Tentative band assignments in the partial Raman spectra of TPU reactive mixture polymerized at $120^{\\circ}\\mathrm{C}$   \n\n\n<html><body><table><tr><td>Raman shift (cm-1)</td><td>Assignment</td></tr><tr><td>2275</td><td>Vassym.(N=C=O)</td></tr><tr><td>1732</td><td>Ester v(C=O), urethane amide I v(C=O)</td></tr><tr><td>1612</td><td>v(Ar)</td></tr><tr><td>1530</td><td>v(Ar), Urethane amide II: v(C-N)+ S(N-H)</td></tr><tr><td>1445</td><td>Vsym.(N=C=O),8(CH)</td></tr><tr><td>1303</td><td>8(CH), urethane amide III?</td></tr><tr><td>1251</td><td>Urethane amide IlI?</td></tr><tr><td>1185</td><td>Urethane amide?</td></tr></table></body></html>  \n\nOf particular interest in Fig. 1 are the asymmetric and symmetric isocyanate stretching vibrations of MDI. Both of these bands noticeably decrease in intensity with polymerization time. Very strong in IR spectra, band intensity of the asymmetric isocyanate stretching vibration is quite weak in Raman spectra. This fact is clearly shown in Fig. 1 where this band is barely discernable from the baseline at a Raman shift of $2275~\\mathrm{{cm}^{-1}}$ . With such a low $\\mathsf{S}/\\mathsf{N}$ , this band could not be used for kinetic analysis. The symmetric isocyanate stretching vibration can be observed as a medium intensity band at approximately $1445~\\mathrm{{cm}^{-1}}$ . Unfortunately, there is considerably overlap of this band with other bands resulting from $\\mathrm{CH}_{2}$ bending vibrations present in all reactants of the TPU formulation. Therefore, this band was not particularly tractable for kinetic analysis either.  \n\nAs shown in Fig. 1, a band at $1530\\mathrm{cm}^{-1}$ is similar to the isocyanate asymmetric and symmetric stretching vibrations in that its intensity decreases with increasing polymerization time. At room temperature, this band is clearly present in both the Raman and IR spectra of pure MDI as shown in  \n\n![](images/d916e6cbbe6944e5bd6c7abed792f99feea172101593ad274e08ef5fe7a6d431.jpg)  \nFig. 2. Raman and IR spectra of pure MDI at room temperature.  \n\nFig. 2. A careful review of the literature suggests that this band arises from para $^{4,4^{\\prime}}$ -isomer) disubstituted phenylene ring vibrations in MDI [17]. Other studies involving band assignment in the Raman and IR spectra of phenyl isocyanate suggest that this band represents one of thirty fundamental $\\scriptstyle{\\mathbf{C}}-{\\mathbf{C}}$ stretching vibrational modes present in the phenyl groups of monosubstituted benzenes [18,19]. Of the 30 fundamental frequencies for $\\mathrm{C_{6}H_{5}–X}$ type molecules, six vibrations are dependent on the mass of X. It is speculated that an analogous vibration is responsible for the $1\\bar{5}30\\mathrm{cm}^{-1}$ band in the phenylene rings of MDI. Regardless, peak intensity of the $15\\bar{3}0\\mathrm{cm}^{-1}$ band, hereon out termed the MDI band, was assumed directly proportional to the concentration of MDI, and thus isocyanate groups, not yet polymerized. Therefore, it was used for determining the polymerization kinetics of this TPU formulation.  \n\nAssuming peak height of the MDI band is a suitable measure of peak intensity, and thus concentration, the relationship between TPU conversion and MDI band peak height can be expressed as  \n\n$$\n\\alpha(t)=\\frac{I_{0}-I(t)}{I_{0}}\n$$  \n\nwhere $\\alpha(t)$ is the time-dependent TPU conversion, $I(t)$ ; the time dependent peak height of the MDI band, and $I_{0}$ is the peak height of this band at zero conversion. Because of the former assumption and the fact that all Raman spectra were normalized with respect to a conversion independent vibrational mode in MDI itself, a method of external calibration was not used in this study.  \n\nThe step growth polymerization of this TPU formulation results in the formation of urethane linkages. These urethane linkages in turn contain $\\mathrm{C-N}$ stretching and $\\mathrm{\\DeltaN-H}$ bending vibrations that are Raman active. Unfortunately, these amide II vibrational modes generate scattering at a Raman shift of approximately $1530\\mathrm{cm}^{-1}$ . Shown in Fig. 1 after $12\\mathrm{h}$ of polymerization time at $120^{\\circ}\\mathrm{C}$ $\\langle\\alpha\\cong1\\rangle$ Þ; these vibrations generate weak yet significant bands in the Raman spectrum of fully polymerized TPU. As a result, Eq. (1) must be modified to account for a growing amide II band at approximately the same Raman shift as the MDI band.  \n\nAssuming peak height of the $1530\\mathrm{cm}^{-1}$ band is a time dependent sum of both MDI and amide II bands and that peak height of the latter band is directly proportionally to conversion, peak height of the MDI band can be written as  \n\n$$\nI(t)=S(t)-A_{\\infty}\\alpha(t)\n$$  \n\nwhere $S(t)$ represents the experimentally measured, timedependent peak height of the composite $1530\\mathrm{cm}^{-1}$ band and $A_{\\infty}$ is peak height of the amide $\\mathrm{II}$ band at complete conversion. Substituting Eq. (2) into Eq. (1) results in  \n\n$$\n\\alpha(t)=\\frac{I_{0}-[S(t)-A_{\\infty}\\alpha(t)]}{I_{0}}\n$$  \n\nAfter realizing that $I_{0}$ equals the composite $1530\\mathrm{cm}^{-1}$ band at zero conversion (ca. 0.40), $S_{0}$ ; and $A_{\\infty}$ equals the composite $1530\\mathrm{cm}^{-1}$ band at complete conversion (ca. 0.12), $S_{\\infty}$ ; Eq. (3) can be rearranged and solved for conversion. This expression  \n\n$$\n\\alpha(t)=\\frac{S_{0}-S(t)}{S_{0}-S_{\\infty}}\n$$  \n\nwas used to calculate TPU conversion data from Raman spectra acquired over the course of an experiment.  \n\nThe symbols in Fig. 3 show conversion versus polymerization time profiles calculated from Eq. (4) applied to chemometrically processed Raman spectra of uncatalyzed TPU reactant mixtures polymerized at various temperatures. As expected, higher polymerization temperatures result in higher conversion rates and final conversions after $30\\mathrm{min}$ of polymerization time. The scattering of data points in Fig. 3 is due to noise in Raman spectra and to errors introduced into the calculation method (i.e. baseline subtraction, normalization, peak height measurement), which are usually inevitable in the quantitative analysis of Raman spectra, especially in kinetic studies.",
        "category": " Materials and methods"
    },
    {
        "id": 11,
        "chunk": "# 3.2. TPU conversion from calorimetry  \n\nAfter completing a temperature scan on a particular sample, TPU conversion versus time data was extracted from the corresponding isothermal scan through application of the proper energy balance. Assuming a constant enthalpic heat of reaction, no significant interference from side reactions, and heat evolved during polymerization was proportional to the extent of polymerization, conversion as a function of time was calculated from  \n\n$$\n\\alpha(t)=\\frac{\\Delta H_{\\mathrm{l}}+\\Delta H(t)}{-(\\Delta H_{\\mathrm{rxn.}})}\n$$  \n\nwhere $\\Delta H_{\\mathrm{l}}$ is the molar enthalpic heat of reaction lost during sample preparation, sample loading, and DSC stabilization, $\\Delta H(t)$ ; the time dependent TPU polymerization exotherm measured in an isothermal scan, and $\\Delta H_{\\mathrm{rxn.}}$ is the total molar enthalpic heat of reaction for TPU step growth polyaddition which was assumed constant for all isothermal polymerization temperatures. $\\Delta H_{\\mathrm{l}}$ was calculated from the relation  \n\n![](images/e502365281a190ff367ad74cd6c2603b0c9bedc2fb293c90da715f5b3973038a.jpg)  \nFig. 3. Experimental and predicted isothermal Raman conversion versus time profiles for the TPU reactant mixture polymerized at different temperatures.  \n\n$$\nH_{1}=-(\\Delta H_{\\mathrm{rxn.}})-\\Delta H_{\\mathrm{t}}-\\Delta H_{\\mathrm{r}}\n$$  \n\nwhere $\\Delta H_{\\mathrm{t}}$ is the total TPU polymerization exotherm measured in an isothermal scan and $\\Delta H_{\\mathrm{r}}$ is the residual molar enthalpic heat of reaction measured in subsequent temperature scan experiments.  \n\nSince TPU step growth polymerization is mixing activated and thus starts with sample preparation, sample loading, and DSC stabilization, DSC was not able to measure exothermic heat flow from the entire course of reaction. Therefore, $\\Delta H_{\\mathrm{rxn}}$ : was measured from independent ATR experiments. In these experiments, the temperature rise of a highly catalyzed, bulk TPU polymerization was followed under quasi-adiabatic conditions. A combination of short total reaction times, fast rate of temperature rise during the major portion of the reaction, and slow heat loss due to low thermal conductivity of the TPU itself ensured that reaction conditions were close to adiabatic. In relating maximum temperature rise to the molar enthalpic heat of reaction for this TPU formulation, $\\Delta H_{\\mathrm{rxn.}}$ ; the following assumptions were made: the TPU reactant mixture was homogeneous, the polymerization was not limited by diffusion, there were no other heat sources other than the polymerization reaction, and density and $\\Delta H_{\\mathrm{rxn.}}$ : were constant. Under these assumptions, the overall energy balance for a single irreversible polymerization reaction, excluding heat loss is  \n\n$$\nC_{p}\\frac{\\mathrm{d}T}{\\mathrm{d}t}=-(\\Delta H_{\\mathrm{rxn.}})\\frac{\\mathrm{d}\\alpha}{\\mathrm{d}t}[\\mathrm{NCO}]_{0}\n$$  \n\nwhere $C_{p}$ is the heat capacity per unit mass, $T$ the experimentally measured temperature, $\\alpha$ the conversion, and $[\\mathrm{NCO}]_{0}$ is initial isocyanate molality.  \n\nIf we eliminate time from both sides of Eq. (7) and assume $\\alpha\\to1$ ; we can integrate the resulting differential equation to solve for $\\Delta H_{\\mathrm{rxn.}}$ as a function of maximum temperature rise. Hence,  \n\n$$\n-(\\Delta H_{\\mathrm{rxn.}})=\\frac{1}{[\\mathrm{NCO}]_{0}}\\int_{T_{0}}^{T_{\\mathrm{f}}}C_{p}(T)\\mathrm{d}T\n$$  \n\nwhere $T_{0}$ and $T_{\\mathrm{f}}$ are initial and final TPU reactant mixture temperatures, respectively. If $C_{p}(T)$ is assumed to be a linear function of temperature and changes very little with conversion from monomer to polymer as is the case for amorphous polymers [20], simple weight average additivity of TPU reactant heat capacities can be used to calculate $C_{p}(T)$ : Using data obtained from Steinle et al. [7], the $C_{p}(T)$ relation used for the TPU reactant mixture formulation in this study was  \n\n$$\nC_{p}(T)=0.9634+0.002776T\n$$  \n\nIn Eq. (9), $T$ is in Kelvin to obtain $C_{p}$ values in $\\mathrm{kJ/kg~K}$ . For the catalyzed TPU formulations investigated here, the average ATR was approximately $67^{\\circ}\\mathrm{C}$ . This leads to an average $\\Delta H_{\\mathrm{rxn.}}$ of $-90\\mathrm{kJ/mol}$ equiv. isocyanate, which is in excellent agreement with the value obtained by other investigators [5 – 9] studying similar systems.  \n\nThe symbols in Fig. 4 show conversion versus polymerization time profiles calculated from Eq. (5) applied to isothermal DSC scans of uncatalyzed TPU reactant mixtures polymerized at various temperatures. Overall, conversion versus polymerization time profiles obtained from calorimetry correlated reasonably well with those obtained from Raman spectroscopy obtained at the same isothermal polymerization temperature (Fig. 3).",
        "category": " Materials and methods"
    },
    {
        "id": 12,
        "chunk": "# 3.3. Kinetic parameter determination  \n\nElemental kinetic mechanisms describing urethane formation from active hydrogen bearing compounds and isocyanates are not well understood. Due to the complexities of urethane reaction mechanisms, most studies have adopted the following Arrhenius type, phenomenological rate law with success [21].  \n\n$$\n\\frac{\\mathrm{d}[\\mathrm{NCO}]}{\\mathrm{d}t}=-k[\\mathrm{NCO}]^{a}[\\mathrm{OH}]^{b}\n$$  \n\nwhere  \n\n$$\nk=A\\ {\\mathrm{e}}^{-E_{\\mathrm{a}}/R T}\n$$  \n\nIn Eq. (10), $k$ is the rate constant and $\\boldsymbol{[\\mathrm{NCO}]}$ and $\\mathrm{[OH]}$ are the concentrations of isocyanate and active hydrogen bearing compounds, respectively. Similarly, the exponents $a$ and $b$ represent the order of reaction with respect to isocyanate and active hydrogen bearing compounds, respectively. As shown in Eq. (11), $k$ is most often expressed with Arrhenius type temperature dependence where $A$ is a frequency factor, $E_{\\mathrm{a}}$ is activation energy, $R$ is the universal gas constant, and $T$ is temperature in Kelvin. It should be noted that Eq. (10) is not a mechanistic model. It has only one rate constant with a single activation energy to express a multitude of reaction mechanisms and rates of reaction.  \n\nIf we assume the urethane reaction is run at equal stoichiometry (i.e. $[\\mathbf{C}]=[\\mathrm{NCO}]=[\\mathrm{OH}])$ ) and express concentration in terms of conversion (i.e.  \n\n![](images/92a94bc12040f8637f38ba0b080da846a7e3d9052f107c10604231837c56cbe2.jpg)  \nFig. 4. Experimental and predicted isothermal DSC conversion versus time profiles for the TPU reactant mixture polymerized at different temperatures.  \n\n$[\\mathbf{C}]=[\\mathbf{C}]_{0}(1-\\alpha))$ , Eq. (10) can be rewritten as  \n\n$$\n{\\frac{\\mathrm{d}\\alpha}{\\mathrm{d}t}}=k[\\mathbf{C}]_{0}^{n-1}(1-\\alpha)^{n}\n$$  \n\nwhere $[\\mathrm{Cl}_{0}$ is equal to initial isocyanate or active hydrogen bearing compound concentration and $n=a+b$ is the overall order of reaction.  \n\nIn order to calculate kinetic parameters for this TPU formulation polymerized at different temperatures, the data in Figs. 3 and 4 was fitted to the kinetic rate law model described by Eqs. (11) and (12) with least squares linear regression techniques. In determining the kinetic parameters $k$ and $n$ from the spectral data, significant data scatter eliminated the possibility of reliably evaluating da=dt: Therefore, the differential equation described by Eq. (12) was solved via the separation of variables technique and rearranged into the following form  \n\n$$\n-\\frac{(1-\\alpha)^{-n+1}}{-n+1}=-\\frac{1}{-n+1}+k[\\mathbb{C}]_{0}^{n-1}t\n$$  \n\nThe left-hand side of Eq. (13) was then calculated from the conversion data in Fig. 3 and plotted versus respective polymerization time for different values of $n$ : The value of $n$ resulting in linear curves, which corresponded to the overall order of reaction, was found to be 1.7 for all four polymerization temperatures investigated. The resulting curves calculated with $n=1.7$ ; which have slopes equal to $k[\\mathbf{C}]_{0}^{n-1}$ and $y$ -intercepts equal to $-1/(-n+1)$ ; are shown in Fig. 5. Using least squares linear regression, each curve was fitted with a regression line and a value of $k$ was found for each polymerization temperature.  \n\nIn determining the kinetic parameters $k$ and $n$ from the calorimetric data, log–log plots of da=dt versus $(1-\\alpha)$ were constructed from the conversion data in Fig. 4 and are shown in Fig. 6. Using least squares linear regression, the linear portion of each curve was fitted with a regression line and a value of $n$ and $k$ was found for each polymerization temperature. Regardless of polymerization temperature, an overall order of reaction of 1.7 afforded the best fit to all the data.  \n\nIt should be noted that only the Arrhenius controlled conversion regime (i.e linear portion) of the curves in Figs. 5 and 6 were fitted with a regression line. Past a critical conversion, which increases with temperature, hard segment phase separation from the reactant mixture results in diffusion controlled kinetics and hence a deviation from the kinetic model described by Eqs. (11) and (12). This is especially evident in Fig. 6 at $100^{\\circ}\\mathrm{C}$ . Such an effect could significantly influence the TPU polymerization exotherm and thus introduce errors into the calculation of conversion from Eq. (5).  \n\n![](images/17a758fdd7ac2950e94131c2a1ee25fb939d5a59e19aaab77d7869a3d8251444.jpg)  \nFig. 5. Plots of Raman $-\\{(1-\\alpha)^{-n+1}\\}/(-n+1)$ versus time for the TPU reactant mixture polymerized at different temperatures.  \n\n![](images/6c71f80f684063d91091054e90dcac24e4f869d6e6262b0dca21282214eb6c68.jpg)  \nFig. 6. log–log plots of DSC conversion rate versus conversion remaining for the TPU reactant mixture polymerized at different temperatures.  \n\nNext, values of $k$ from the linear regression analyses in Figs. 5 and 6 were used to calculate the Arrhenius kinetic parameters $A$ and $E_{\\mathrm{a}}$ from both the spectral and calorimetric data, respectively. Assuming the kinetic rate law model given by Eq. (12) is valid and $n$ remains constant throughout the entire reaction, a semi-ln plot of $k$ versus $1/T$ should yield a straight line with slope equal to $-E_{\\mathrm{a}}/R$ and $y.$ - intercept equal to ln A : Indeed, as shown in Fig. 7, values of $k$ obtained from the two sets of data do form linear curves when plotted versus $1/T$ and almost coincide showing that the two different measurement techniques yielded similar results. The Arrhenius parameters $A$ and $E_{\\mathrm{a}}$ were evaluated from the least squares linear regression lines also shown in Fig. 7. Table 2 lists all kinetic parameters calculated for the step growth polymerization of this TPU formulation and Figs. 3 and 4 show the corresponding model predictions.  \n\n![](images/025852f3a7ed036ff4545b3f6703156fcff4159c1622deecc6b1378fb8454b12.jpg)  \nFig. 7. Evaluation of $A$ and $E_{\\mathrm{a}}/R$ from semi-ln plot of Raman and DSC $k$ versus $1/T$ data for TPU reactant mixtures polymerized at different temperatures.  \n\nTable 2 Listing of kinetic parameters obtained from TPU investigated in this study ${\\bf d}[{\\bf C}]/{\\bf d}t=-A\\ {\\bf e}^{-E_{\\mathrm{a}}/R T}[{\\bf C}]^{n}$   \n\n\n<html><body><table><tr><td>Characterization technique</td><td>A (mole NCO-0.7/kg solution-0.7 s)</td><td>Ea (J/mol)</td><td>n</td></tr><tr><td>Raman</td><td>6.02 × 102 ± 1.62·102</td><td>3.87 × 104 ± 5.73 × 103</td><td>1.7</td></tr><tr><td>DSC</td><td>9.92 × 10² ± 1.18 × 102</td><td>3.87 × 104 ± 2.74 × 103</td><td>1.7</td></tr></table></body></html>\n\nNote: $[\\mathrm{C}]{=}[\\mathrm{NCO}]{=}[\\mathrm{OH}]$ has units of mole/kg.  \n\nIn summary, both Raman spectroscopy and DSC yielded similar results when used to measure the kinetics of TPU polymerization. Both measurement techniques yielded 1.7 for the overall order of reaction, $n$ : This is in agreement with almost all urethane reaction kinetic data in the literature, where $n$ varies from 1 to 2 [9,21]. For example, using isothermal and non-isothermal DSC, Hager et al. [3] calculated $n$ to be 2.0 while Hernandez-Sanchez and VeraGraziano [4] calculated $n$ to be 1.63, respectively. Utilizing the ATR measurement technique, Lipshitz and Macosko [5] calculated $n$ to be 1.5 while Camargo et al. [8] calculated $n$ to be 1.4.  \n\nThe activation energy, $E_{\\mathrm{a}}$ ; calculated from both measurement techniques was approximately $3.9\\times10^{4}\\mathrm{J/mol}$ , which is general agreement with the literature. For example, Hager et al. [3] calculated $E_{\\mathrm{a}}$ to be $4.1\\times10^{4}\\mathrm{J/mol}$ with isothermal DSC and Camargo [9] calculated $E_{\\mathrm{a}}$ to be approximately be $5.5\\times10^{4}\\mathrm{J/mol}$ with ATR. Incidentally, values of $E_{\\mathrm{a}}$ closer to that obtained by Camargo [9] are obtained from both measurement techniques if the $100^{\\circ}\\mathrm{C}$ data in Fig. 7 is not included. As previously discussed, it is probable that micro-phase separation results in a significant deviation from Arrhenius controlled reaction kinetics at this temperature. A deviation from the kinetic model described by Eqs. (11) and (12) caused by urethane bond thermal dissociation is also possible. Several publications [22–24] have shown that this process starts at $150-160^{\\circ}\\mathrm{C}$ and becomes significant at $190-200\\ {^{\\circ}}\\mathrm{C}$ . In this study, it is assumed that such depolymerization is insignificant at all of the temperatures investigated. Given the linearity of the data in Fig. 7 at 120, 140, and $160^{\\circ}\\mathrm{C}$ , this assumption appears reasonable.  \n\nThe frequency factor, $A$ ; obtained from DSC is approximately $50\\%$ larger than that obtained from Raman spectroscopy. While both are in general agreement with literature values for uncatayzed TPU systems [3], this disparity probably results from errors introduced into the calculation of conversion (i.e. baseline subtraction and normalization in Raman spectroscopy and the calculation of $\\Delta H_{\\mathrm{rxn.}}$ in DSC) from both measurement techniques. Another probable source for this difference comes from the fact that Raman spectroscopy is a direct (i.e. measures reactant concentration) technique while DSC is an indirect (i.e. measures heat evolution) technique. Differences in measurement technique sample geometry and preparation could also be significant. Regardless, kinetic parameters obtained from both measurement techniques agreed favorably with classical literature values [3–9] proving that Raman spectroscopy is a useful method for characterizing the kinetics of polyurethane polymerization.",
        "category": " Results and discussion"
    },
    {
        "id": 13,
        "chunk": "# 4. Conclusions  \n\nPeak intensity of a band in MDI was assumed proportional to isocyanate concentration and thus conversion in the polymerization of this TPU formulation. Given the capability to calculate conversion from Raman spectra acquired over the course of an experiment, conversion versus time data was collected from the isothermal polymerization of this TPU formulation at different temperatures. Such data was modeled with an Arrhenius type, phenomenological rate law with success. Kinetic parameters agreed reasonably well with those obtained from analogous calorimetric measurements and with literature values. Since the Raman effect is a scattering process, sample preparation is relatively simple compared to other spectroscopic techniques. Hence, it can be concluded that Raman spectroscopy is a powerful tool for characterizing the polymerization kinetics of polyurethanes in situ.",
        "category": " Conclusions"
    },
    {
        "id": 14,
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        "category": " References"
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]