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Anal. Chem. 1989, 6 7 , 1202-1207
Theoretical Model for Quantitative Determination of Volatile Compounds in Polymers by Dynamic Headspace Sampling Anders Hagman*J
Department of Analytical Chemistry, Arrhenius Laboratory, University of Stockholm, S-106 91 Stockholm, Sweden Sven Jacobsson
K A B I Pharma, Research and Development Department, Box 1828, S-171 26 Solna, Sweden
A theoretical model was derived for dynamic headspace Sampling of voiatiies in solid samples, such as polymers, and in which process parameters as well as component parameters were included. The model assumes that the mass transfer from the polymer to the surrounding gas phase is accomplished by two rate constants, one wlthin the polymer and one in the boundary gas layer at the polymer interface. By use of the model the recovery can be determined, which simplifies the quantlflcation. The calculated and the experimental results show very good correlation under different types of conditions. The effects of variation of some process parameters (e.g. temperature and flow rate) on the recovery were also studied. Under the studied temperature range, as hlgh a temperature as possible, but lower than the melt tndex for the polymer, gives the best possibility to achieve highest recovery for less volatlie compounds. However, the best solution for all types of compounds is to have a small film thickness and as high a contact area as possible. A decisive parameter is the thickness of the boundary layer, which should be mlnlmized as far as possible to achieve rapid transport in the gas phase. The mass transfer through this layer is often the rate-determining factor for the extraction efficiency. The flow rate optimum is determined by the geometry of the headspace sampler, type of extraction gas, and the cold trap efficiency.
INTRODUCTION Dynamic headspace (DHS) is a technique in which the sample is constantly swept by a stream of an inert gas, volatile compounds being thus removed from the sample matrix and transferred to some kind of collection device, e.g. a cold trap. This technique is versatile and can be applied to the analysis of volatiles in different matrices, such as analysis of additives in polymers ( I , 2 ) and organic pollutants in water (3). DHS has been used primarily for qualitative analysis, and quantitative results are scarce in the literature (4-6). Quantification can be carried out by repetitive extraction cycles until a satisfying amount has been released for an accurate determination (7). However, each desorption/trapping and analysis step is often quite time-consuming, so it would be more convenient to be able to predict the recovery and thus reduce the number of extraction cycles. Evaluation of a theoretical model for prediction of the recovery from dynamic headspace extraction of liquid samples has been demonstrated by Curvers et al. (8). With regard to solid samples, the diffusion process within the sample is more significant and more complex than for liquids. Therefore a
* Author to whom correspondence should be addressed.
Present address: KABI Pharma, Research and Development Department, Box 1828, S-171 26 Solna, Sweden. 0003-2700/89/036 1-1202$01.50/0
mathematical model must be based on several component/ matrix properties, such as distribution coefficient, diffusion coefficient, and mass transfer coefficient. Mathematical models, including these component/matrix properties, for different diffusion problems have been described by Crank (9, l o ) ,but none of the models could be totally applicable for the dynamic headspace sampling. Another feature with a model is that the desorption parameters (e.g. flow rate, temperature) can be optimized in such a way that they will achieve more effective extraction. This study shows the effects of various DHS process parameters as well as component properties (distribution coefficient, diffusion coefficient, and mass transfer coefficient) on the recovery of some compounds in a polypropylene material. Process parameters include flow rate of the extraction gas, desorption temperature and time, and film thickness of the polymer. The experimental results are compared to those generated when a theoretical model is used. EXPERIMENTAL SECTION The dynamic headspace system used in this study was designed and assembled in our laboratory (11, 12). The sample was a polypropylene material in the form of granules that were cut down to slices of different thickness by a microtome. A polymer sheet (30 mg) was then placed in a perforated sample holder of quartz (11.5 cm X 6 mm 0.d. X 4 mm id.), which was in turn placed in the dynamic headspace chamber (DHC). The DHC consisted of a quartz tube (12.4 cm X 9 mm o.d. x 6.9 mm i.d.), the diameter of which was gradually reduced toward the end of the tube in the flow direction at a rate of 1.1 mm per millimeter length of the tube to a diameter of 3.2 mm o.d. During the desorption mode, the DHC was heated and a gas flow of helium was passed over the sample, by which the desorbed components were transferred to a cold trap. The cold trap was made of deactivated fused silica (30 cm X 0.53 mm i.d.) filled with deactivated glass beads (80-100 mesh). This fused silica tube was placed inside a U-shaped glass tube (2.0 mm 0.d. X 1.2 mm id.), which was suspended in a Dewar flask containing liquid nitrogen. Around the glass tube a Kantahal A wire was coiled to achieve rapid heating at injection. The collected compounds were injected into a Packard 428 gas chromatograph equipped with a SE-54 capillary column (20 m X 0.22 mm i.d.). The column, which was programmed from 50 "C (2 min) to 280 "C at 10 "C/min, was connected to a flame ionization detector. To determine the total amounts of the studied compound in the polypropylene material, the described procedure was repeated for the same sample as many times as was necessary to achieve accurate quantitative determination. Three different compounds, hexane, tridecane, and 2,6-ditert-butyl-4-methylphenol (BHT), were studied. All data treatment was carried out on an Olivetti M24 personal computer, and the program was written in QUICK-BASIC (Microsoft). Dynamic headspace process parameters studied were as follows: Flow Rate of the Helium Purge Gas. To study the effect of the flow on the yield, six different flow rates, 5, 10, 15, 20, 25, and 30 mL/min, were studied under different conditions. The 'C 1989 American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 61, NO. 11, JUNE 1, 1989
solid phase -I= Ras phase
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Figure 1. Depiction of the mass transfer system in relation to the theoretical model: J,, mass flux within the polymer (mol/s cm'); J,, mass flux over the polymer interface and at the boundary gas layer (mol/s cm2); K , distributioncoefflclent; F , flow rate of the gas (cm3/s); CEO,initiii concentration in the polymer (mol/cm3); C,,,concentration in the solid side of the polymer interface; C,, concentration in the gas side of the polymer interface; C* concentration at the convective gas layer; C,(x J ) , concentration profile within the polymer; C,(t), concentration profile In the gas phase; L , half-thickness of the polymer sheet (cm); a,, fictitious film thickness of the laminar flow (cm); T,, thickness of the convective, turbulent flow (cm).
flow rate was controlled by a flow controller (Porter Instrument Co., Hatfield, U.K.) and a thermal mass flow sensor (F-lll-EA, Bronkhorst high-tech B.V.), which registers the flow rate momentarily but also gives information about the total purge volume. Likewise, an internal standard system based upon continuous generation of gas standard mixtures (12) also gave an accurate determination of the purge gas volume. Temperature of the Dynamic Headspace Chamber. In order to study the effect of the temperature on the yield, six temperature levels were chosen: 95,115, 130,150,165, and 170 OC. The melt index of the polymer was 165 OC. The temperature was regulated by a temperature regulator (Shinko, Electronic Instrument Co., Ltd., Osaka, Japan) and controlled by a digital thermometer (Keithley Instruments, Inc., Cleveland, OH). Desorption Time. To study the influence of the desorption time on the recovery, five different desorption times were used: 5, 10, 15, 20, and 25 min. The interval of the desorption mode was controlled by an electronic timer (Velleman N.V., Gavere, Belgium). Sample Thickness. Six different film thicknesses of the polymer sample, 50,100,200,300,400,and 500 pm, were examined for the relationship with the yield. By use of a microtome (Leitz, Wetzler, FRG) a thin, reproducible cut could be made at a fixed position on the polymer granule. THEORETICAL SECTION In this study, a mathematical model was derived for dynamic headspace extraction in order to predict the recovery and with which the use of repeated desorption cycles can be reduced or even eliminated. Another feature of the model is that a relationship between the desorption parameters (e.g. temperature and flow rate) and the component/polymer properties (e.g. the diffusion coefficient and the distribution coefficient) will be obtained. This model, which follows a slightly different derivation than that given by Gandek and Hatton (13) and Hatton et al. (14),is based on a mass transfer situation, illustrated in Figure 1. A sheet of polymer with a thickness of 2L contains at time zero a uniform concentration of a volatile compound. Both sides of the sheet are in contact with the gas phase. As the volatile compound at the surface will evaporate, the concentration within the polymer will be changed with time and position ( C ( x , t ) ) . Therefore, a mass flux of molecules will take place within the polymer (J,)and As the flow another mass flux over the polymer interface (Ji). of helium comes into contact with the solid, a layer of gas, which is free from mixing by convection, is formed at the interface. This layer, called the boundary layer, has a laminar flow, and any mass transfer through this layer is accomplished by molecular diffusion. A difference from the model given by Gandek and Hatton (13) is that the mass transport from
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the polymer surface to the gas phase (i.e. evaporation) is considered to be very fast compared to the transport through the boundary layer and has therefore not been taken into consideration. Thus, the equilibrium between the solid and the gas phase at the polymer interface will be quickly obtained. Another difference is that the gas volume, which is sampled up to time t , is considered to be constant (Le. the flow rate of the extraction gas does not change with time). In another region of the gas phase the volatile compounds undergo very rapid diffusion and convection, due to the turbulent flow. According to the Reynolds number (a dimensionless term describing the properties of the flow), the flow through the sampler tube should be considered as a laminar flow and not as a turbulent flow. However, in this case a perforated sample holder is placed in the tube, which makes the flow less well defined than that in an open tube. The flow is more turbulent-like due to disturbance of the laminar flow by stream splitting. The mass flux for a compound within the polymer is assumed to be described by Fick's second law
and is dependent on the diffusion coefficient (D) and the concentration gradient. The mass flux at the interface and the boundary layer is given by and is dependent on the mass transfer coefficient (k,) for the boundary layer and the concentration difference (C,i - C,) over the boundary layer. The model is based on the following assumptions: (a) The polymer sheet initially contains a uniform concentration of the solute. It is worth noticing that this is never totally achieved because of compounds that migrate to the surface by repelling interaction between the compound and the polymer itself and volatile substances that evaporate to the ambient air during storage. (b) Both sides of the polymer sheet are in contact with the movable gas phase. The edges of the sheet have a negligible effect. (c) Migrant diffusion within the polymer follows Fick's laws. Non-Fickian and case I1 diffusion behavior differ from Fickian by the fact that the diffusion is rapid compared to the polymer relaxation process (15). Case I1 diffusion is pronounced a t high concentrations of solutes in the polymer (causes the polymer to swell, and the free volume in the matrix increases, resulting in a higher diffusion rate) and at temperatures near or below the glass transition temperature of the polymer (decreases the relaxation rate of the polymer). Non-Fickian or "anomalous" behavior lies between case I (Fickian) and case I1 diffusion and changes often sigmoidally from one to the other. However, in this study the temperature (95-170 "C) was well above glass transition temperature, and the total concentration of the volatile substances was low (
