w Correction and Reassessment of Intrinsic Viscosity and The Universal Calibration Method - A Bridge from Polymers to Small Molecules Advanced TechnologyShyhchang GroupS. Hua,ngBFGoodrich I Breckn, ilIe, OH 44141 '_ Abstract: The current definition of intrinsic viscosity ([riD, and its application in the universal calibration method, applies to polymeric materials with molecular weight (MW) >>1,000. However, there have been numerous discussions regarding the [11]of low MW oligomers. Difficulties can be encountered at very low MW, because the [1"1]in some cases becomes negative, which in theory is impossible. A new model is proposed in this paper to refine the calculation of [rl]. The new model is based on the fact that [ri] is a polymer volume expansion coefficient; i.e., [11]times the polymer density is the expansion ratio of the hydrodynamic volume in solution to the polymer volume in the solid state. In the calculation of [rl], as in measuring the density of a low-density material in air, the volume of solvent should be added. For a polymer with MW greater than 10,000, this correction can be neglected. However, the correction becomes important for small oligomers. With this correction, the Mark-Houwink curve of n-alkane and polystyrene in the low MW region become linear. The universal calibration curve, calculated with corrected [rl], is also linear across the range from polymer to small molecules. INTRODUCTION In recent years, there has been increasing interest in producing low molecular weight (MW) polymers and oligomers with applications as plasticizers, detergents, food additives, lubricant additives, copolymer precursors, et.aL. Thus the determination of accurate and/or absolute MW distribution and MW averages becomes increasingly more important for the purposes of quality assurance, research and development, patent applications, and product litigation. However, characterization of low-molecular weight polymers is difficult for both fundamental and practical reasons. One example is that several parameters for polymers, such as refractive index and density, do not remain constant in the low oligomer range. Size-exclusion chromatography (SEC) is by far the most common method for MW distribution studies of polymers and oligomers. The retention volume in SEC is based on the polymer size in the mobile phase. The MW is calibrated with a set of known standards or is determined directly using an in-line MW detector, such as a light scattering photometer, viscosity detector, or mass spectrometer. In 1967 Benoit et.al, introduced a universal calibration method in which the retention of a molecule in SEC is related to its hydrodynamic volume. This can be calculated by multiplying the MW by the intrinsic viscosity ([rl]) t_]. The [rl] of a polymer is related to its MW by the equation: [rfl = kM" (1) where M is the MW of polymer, k and a are Mark-Houwink constants. Therefore, a universal calibration curve for an SEC column set can be established using a set of polymer standards having a wide MW range with known [1"1]values. The true MW distribution of other C- C 328 the idea of segment concentration and plotted [_]]_ versus MW _/2. This method appeared to be particularly useful in the low and medium MW range. A linear relationship of polystyrene in benzene was found down to MW 2,500 t2). To correct the deviation of low MW material, Dondos and Skordilis proposed an error term related to the exponent a of the Mark-Houwink- Sakurada equation c3). Yamakawa and his colleagues attempted to use the exclusion-volume effect and an empirical parameter to treat the negative [rl] of low oligomers, ta-s)but was not successful. Recently, Chance and his colleagues also investigated the [rl] of low MW oligomers. The again found that universal calibration method failed in the low oligomer region t6). In this paper the problem is looked at from a different perspective, and a new model is proposed to correct the [rl] of polymers and oligomers. The corrected [1"1]is defined as "{rI}" (using curly rather than square brackets). After correction, the log {rI} versus log(MW) for low MW oligomers of n-alkane and polystyrene is linear, indicating that {rl } will not become negative. The universal calibration curve calculated with {rl} can smoothly connects high polymers with small molecules. i THEORY What is Intrinsic Viscosity The intrinsic viscosity, [rl], or dilute solution viscosity, is measured using a capillary viscometer, where: [rl] = ( %/c (2) in which, rlspis called specific viscosity where: rl,p = rl,-1 =t/t 0- 1 (3) where Tit is the relative viscosity (t/t0), t is the afflux time required for a specified volume of polymer solution to flow through a capillary tube, and to is the corresponding afflux time of the solvent, rl_p(equation 3) is the excess viscosity contributed by polymer, i.e., in addition to the viscosity from the solvent. Therefore, the [rl] is mainly a measurement of the "change" due to the polymer when it is dissolved in solvent. As early as 1930, Staudinger and Heuer r_ attempted to relate this viscosity number to the molecular weight of the polymer. In 1938, Mark is) demonstrated this relationship empirically as equation 1, commonly called Mark- Houwink equation, or Mark-Houwink-Sakurada equation. Flory also investigated the relationship between [rl], MW, and the molecular size in solution tg]. However, the physical significance of [rl] is still not well understood. 329 V i , Since the universal calibration method was introduced, its applicability has been proven for high polymers. According to the theory of the universal calibration methou,tlj , Vha = [filM (4) where Vhd is the hydrodynamic volume of polymer in solution. For the polymer in the solid state, V_d = M/p, (5) or M = Vsdp (6) where V_ and p are the volume and density of polymer in solid phase, respectively. Combining equations 4 and 6: Vhd = [rl]V,dp, (7) therefore, [11] = (V_ / Vsd)(1/p) (8) or, []lip _'_ Vhd / V_I (9) In other words, the intrinsic viscosity is actually a comparison of the hydrodynamic volume of a polymer in solution to its original volume in the solid state. The product of the intrinsic viscosity and the density is the expansion ratio. Table 1 shows data for polystyrene. The density of polystyrene is approximately 1.0 g/mL and the [11]of polystyrene (MW 3x105) in tetrahydrofuran is approximately 100 mL/g. Therefore, the polymer chain expands 100 times in solution. Likewise, the expansion ratio of polystyrene (MW 8x106) is approximately 1,000. Because the expansion ratio increases with MW very rapidly, the high MW polymer chains are easily deformed and degraded under shear. The higher the MW, the more severe the deformation as well as degradation. i Correction of Intrinsic Viscosity it I For high molecular weight materials for which the universal calibration method applies, the above relationship will also apply. However, for low molecular weight oligomers, this relationship breaks down. At very low MW, [1"1]becomes negative. This is incorrect, since neither Vhd or V_dCan be negative. Why is the intrinsic viscosity negative? Let us recall equations 2 and 3 for the calculation of the intrinsic viscosity. The [rl] becomes negative when the t for a low oligomer or small molecule is less than t0. As discussed previously, the intrinsic viscosity is a measurement of volume change. When the hydrodynamic volume of a polymer is much greater than the volume of a solvent molecule, t will be greater than to. However, when the hydrodynamic volume of a small oligomer is the same as the solvent molecule, there will be no net increase in the viscosity. The t will be the same as t0, and the apparent [1"1]becomes zero. When the hydrodynamic volume of the oligomer is smaller than the solvent molecule, the intrinsic viscosity becomes negative. This is similar to measuring the density of a material in air. If the density of the material is much greater than air, the calculated density will be close to the correct value, when measuring the density of a gas that is lighter than air, the directly calculated "density" becomes negative. The way to correctly calculate the 330 ........... ._._._..... :--'.,,_'__:._...z_ - , ._ ..___ - _.._. _,.._- -._,_ density of a gas is to add the weight of air of the same volume to the measured weight of the gas. Likewise, to "correct" the [rl] of a low oligomer one may add the hydrodynamic volume of the solvent molecule to the apparent hydrodynamic volume of the oligomer, [rl]Mp. According to this hypothesis, we can define a "corrected intrinsic viscosity", {r]}, which takes into account the volume of the solvent molecule: Vhd -- {rl}Mp = [rl]Mp + V,v* (10) in which Vsv* is the hydrodynamic volume of the solvent molecule. The "*" is added because the V_vneeds to be adjusted. During the calculation of [_], rl_, is divided by the concentration of the polymer or oligomer, in units of weight per volume. Therefore, the [rl] is based on the polymer (or oligomer) density. Every term in equation 10 needs to be multiplied by its own density: Therefore, {1]} = [1]] +Ms,/MI, p p (12) For high polymers with MW > lxl04, the MJMppp term is negligible. However, for low oligomers, this correction becomes significant. Hydrodynamic Volume of Small Molecules If the above correction is correct, then equation 12 should also be applicable to small molecules, and the universal calibration curve of polymers and oligomers calculated using that equation should be extendable to small molecules. However, the intrinsic viscosities of small molecules are normally very low and are difficult to measure accurately.