Intrinsic viscosity, represented as \([\eta]\), is a specific measure in polymer science that quantifies a single polymer molecule’s ability to increase the viscosity of a solvent. This value represents the polymer’s contribution to the solution’s viscosity at a state of infinite dilution, where individual polymer chains no longer interact. Calculating \([\eta]\) is a primary method for characterizing large molecules, offering insights into their size, shape, and molecular weight. It connects the macroscopic measurement of fluid flow with the microscopic properties of the dissolved polymer chain.
Defining the Intermediate Viscosity Values
The calculation of intrinsic viscosity requires determining several intermediate viscosity terms from the raw experimental data. The first is Relative Viscosity (\(\eta_r\)), the ratio of the polymer solution’s viscosity (\(\eta\)) to the pure solvent’s viscosity (\(\eta_0\)). In capillary viscometry, \(\eta_r\) is equivalent to the ratio of the solution flow time (\(t\)) to the solvent flow time (\(t_0\)).
The Specific Viscosity (\(\eta_{sp}\)) isolates the increase in viscosity solely attributed to the presence of the polymer solute. This is calculated by subtracting one from the relative viscosity: \(\eta_{sp} = \eta_r – 1\). This dimensionless quantity measures the fractional increase in viscosity caused by the polymer.
Two further intermediate values depend on the polymer’s concentration (\(c\)). Reduced Viscosity (\(\eta_{red}\)) is found by dividing the specific viscosity by the concentration: \(\eta_{red} = \eta_{sp}/c\). This term reflects the efficiency with which the polymer contributes to viscosity per unit of mass.
Inherent Viscosity (\(\eta_{inh}\)) is calculated as the natural logarithm of the relative viscosity divided by the concentration: \(\eta_{inh} = \ln(\eta_r)/c\). Both the reduced and inherent viscosities share the same units as intrinsic viscosity, typically deciliters per gram (dL/g), and serve as the starting points for the final extrapolation.
Practical Steps for Measurement
Obtaining the raw data necessary for these calculations involves precise experimental work, typically using a capillary viscometer. The Ubbelohde viscometer is commonly employed because its suspended-level design ensures the pressure driving the fluid flow is independent of the total sample volume. The viscometer must be immersed in a constant temperature bath to maintain thermal equilibrium, as viscosity is highly sensitive to temperature fluctuations.
The core of the measurement involves preparing a series of dilute polymer solutions at several known, decreasing concentrations (\(c\)). The flow time (\(t\)) is measured for each solution as it passes between two calibrated marks on the viscometer’s capillary tube. This process is repeated to ensure high precision, and an average flow time is recorded.
The flow time of the pure solvent (\(t_0\)) must also be measured under the exact same temperature conditions. The ratio of the solution flow time to the solvent flow time provides the relative viscosity (\(\eta_r\)), the foundational data point. Using very dilute solutions (where \(\eta_r\) is typically between 1.2 and 2.5) minimizes inter-chain polymer interactions that could skew the results.
Extrapolation Methods for Final Calculation
Intrinsic viscosity is not a directly measured quantity but is determined by extrapolating the concentration-dependent intermediate values to zero concentration (\(c \rightarrow 0\)). This extrapolation removes the influence of polymer chains interacting with each other, leaving only the influence of a single, isolated chain. The two primary mathematical models used for this linear extrapolation are the Huggins and Kraemer equations.
The Huggins equation relates the reduced viscosity to the concentration via the formula \(\eta_{red} = [\eta] + k_H [\eta]^2 c\). To apply this, the reduced viscosity (\(\eta_{sp}/c\)) for each solution is plotted against its corresponding concentration (\(c\)). The resulting data points should fall on a straight line in the dilute regime, and the y-intercept of this line yields the intrinsic viscosity \([\eta]\).
The Kraemer equation, also known as the Mead-Fuoss equation, uses the inherent viscosity to perform a similar extrapolation: \(\eta_{inh} = [\eta] – k_K [\eta]^2 c\). Here, the inherent viscosity (\(\ln(\eta_r)/c\)) is plotted against the concentration (\(c\)). The y-intercept of this second plot should, in theory, yield the identical intrinsic viscosity value. This dual plotting method provides a check on the experimental data’s reliability, as both extrapolated lines must converge at the same y-intercept.
The constant \(k_H\) in the Huggins equation, known as the Huggins constant, provides a measure of the polymer-solvent interaction and the extent of chain-chain entanglement. For ideal solutions, \(k_H\) typically falls within a narrow range, often between 0.3 and 0.5. A value closer to \(0.3\) suggests a “good” solvent where the polymer chain is highly expanded, while a value closer to \(0.5\) indicates a “poor” solvent where the chain is more tightly coiled.
Applying Intrinsic Viscosity to Polymer Characterization
The final, calculated intrinsic viscosity value provides a direct link to the physical characteristics of the polymer chain. The most important application is the determination of the polymer’s viscosity-average molar mass (\(M_v\)). This is achieved using the empirical Mark-Houwink equation, which establishes a power-law relationship between intrinsic viscosity and molar mass: \([\eta] = K M_v^a\).
The parameters \(K\) and \(a\) in this equation are constants that are unique to a specific polymer, solvent, and temperature combination. These constants must be determined experimentally or obtained from reference tables for the system being studied. Once the constants are known, the measured intrinsic viscosity allows for a straightforward calculation of the polymer’s molar mass.
Beyond molar mass, the exponent \(a\) in the Mark-Houwink equation provides valuable information about the conformation, or shape, of the polymer chain in the solution. Values of \(a\) can range from approximately 0.5 for a flexible, randomly coiled chain in an ideal solvent (a theta solvent) to up to 2.0 for a stiff, rod-like polymer structure. The intrinsic viscosity measurement offers a method for understanding how a polymer behaves in a given environment.