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An Experimental Review: Evaluation of the Flory-Fox Equation for the Relationship of Glass Transition Temperature (Tg) vs Molar Mass of Polystyrene Using Differential Scanning Calorimetry (DSC)

Ronald P. D’Amelia*, Brandon Khanyan

Chemistry Department, Hofstra University, Hempstead, NY

Abstract: Differential Scanning Calorimetry (DSC) is a technique which measures heat flow to study phase transitions, thermodynamic properties, and oxidation reactions. One particular use of the technique is to observe glass transition temperatures of polymers by measuring the energy changes upon heating. The experiment entails the measurement of glass transition temperatures and enthalpies of transition for monodispersed polystyrene (PS) samples, as well as the analysis of binary mixtures ranging from 0% to 100% in 20% intervals of monodispersed polystyrene of various molecular weights. The results corroborate the molecular weight dependent of glass transition temperature curve for monodispersed polystyrene as predicted by the Flory-Fox equation, Tg = Tg,∞ - K / Mn. The second goal of the experiment is to determine the weight percent proportion in each of the mixtures studied. The experimental results for the polystyrene mixtures show a strong, linear correlation between the theoretical and experimental glass transition temperatures found using DSC. The experiment is designed to demonstrate the applicability of the Flory-Fox equation and to increase awareness of the applications of DSC that could be integrated into undergraduate analytical and instrumental polymer chemistry laboratory course curriculums.

Keywords: Differential Scanning Calorimetry, polystyrene, binary polystyrene mixtures, Flory-Fox equation, Glass transition temperature Tg , undergraduate laboratory experiment

Received July 09, 2022; Revised August 15, 2022; Accepted August 23, 2022

Citation: Ronald P. D’Amelia, Brandon Khanyan, An Experimental Review: Evaluation of the Flory-Fox Equation for the Relationship of Glass Transition Temperature (Tg) vs Molar Mass of Polystyrene Using Differential Scanning Calorimetry (DSC). Journal of Polymer and Biopolymer Physics Chemistry, 10 (1). 10-17. January 2022. doi:10.12691/jpbpc-10-1-2

1. Introduction

Polystyrene is one of the most used plastics in the world and is commonly found in protective packaging, insulation and cushioning [1,2,3,4]. The structure contains phenyl groups off alternating carbon centers of variable length. Figure 1 shows a styrene monomer of polystyrene, where n represents the number of backbone carbon chain repeats.

The Flory-Fox equation predicts the relationship between the molecular weight and the glass transition temperature of a given polymer [5,6,7]. The glass transition refers to the change from a rigid, glass material to a softer flexible one. The glass transition temperature (Tg) marks the point at which the transition occurs, characterized by a sudden change in heat capacity and thermal expansion. Using differential scanning calorimetry (DSC), the change in heat capacity can be measured to determine Tg. This experiment determines the glass transition temperature of monodispersed polystyrene with different molecular weights, corroborating the Flory-Fox equation. The glass transition temperature of a material initially increases as molecular weight increases and then levels off after a certain molecular weight. DSC results were obtained by running all the samples against an empty reference pan. The Tg of the polystyrene samples were determined using the Perkin-Elmer DSC thermal analysis software Pyris for Windows.

Figure 1. Structure of polystyrene monomer

Binary mixtures ranging from 0 to 100 wt. % in 20% intervals of monodispersed polystyrenes with different molecular weights were prepared and analyzed. This experimental review has been designed to implement DSC methodology and analysis into undergraduate polymer chemistry laboratory curriculums.

2. Background Perspective [8-13]

A solid polymer can be distinguished into the amorphous and semi-crystalline categories. Amorphous solid polymers are either in the glassy state, or in the soft and rubbery or fluid state. The typical model of a macromolecule in the amorphous state is the “random coil”, however, the amorphous state is better depicted by an irregularly folded chain molecule rather than a completely idealized random coil.

The glassy state of a polymer can be described as a state in which cooperative chain motion of the macromolecules are frozen, such that only limited local motion can take place, such as side-group rotational/vibrational motion. These motions exist due to bond angle deformation and bond stretching within the molecules.

2.1. Glass Transition and Glass Transition Temperature

One of the most important properties of both amorphous and semi-crystalline polymers is their thermal behavior. Understanding this behavior is not only critical for the selection of proper processing and manufacturing conditions, but also for the full understanding of the polymeric physical and mechanical properties.

The most prevalent transition in amorphous polymers, is usually labelled the vitreous or glass transition, in which the linear or volume coefficient of thermal expansion increases. In contrast, in semi-crystalline polymers, the glass transition usually occurs below the melting temperature. The exact description of the molecular motion responsible for the glass transition is undefined, however, it is generally thought to involve macromolecular random chain bond movement, such as groups or segments of the polymer macromolecular relaxing, vibrating or reptating (crawling). Above the Tg, the chain segments can undergo cooperative rotational, vibrational, translational, and diffusional motion. The importance of the glass transition in polymer science was stated by Eisenberg: “The glass transition is perhaps the most important single parameter that determines the application of many non-crystalline polymers now available” [14].

The temperature-dependent properties of amorphous polymers undergo major changes at the glass transition temperature (Tg). The simplest of many definitions of the glass transition temperature (Tg) is the temperature below which the amorphous polymer is glassy, and above which is soft and rubbery. The molecular interpretation of Tg is the temperature of the onset of large-scale motion of molecular chain segments. Below the Tg, the polymer chains’ atoms undergo little rotational-vibrational motions or are in a frozen bulk solid state.

The glass transition temperature can be measured in a variety of ways, not all of which yield the same value. The results from the kinetic and thermodynamic nature of the glass transition differ, and the Tg is dependent upon the thermal history of the polymer and the heating/cooling rate of the experiment.

2.2. Theories for Glass Transition (Glassy -to- Rubbery)

Many theories regarding the glass transition have been previously developed. They include: the iso-free volume theory by Flory-Fox [5,6,7], a modified mechanical-free volume phenomenological theory that was explored by the Williams-Landel-Ferry (WLF) equation [15], a modified Free volume Relaxation-Kinetic theory that includes Lattice-Hole/Voids developed by Hirai-Eyring [16], and finally, the Gibbs-DiMarzio [17,18] thermodynamic theory which suggests that the transition is a true second order thermodynamic transition representing an equilibrium between the glass and rubbery state in which the conformational/configurational entropy at equilibrium is zero.

2.3. Free Volume Theory of Glass Transition

The Flory-Fox Iso-Free Volume theory postulates that the glass transition occurs when the free or unoccupied volume in the macromolecule reaches a constant value and does not decrease further as the polymer is cooled below or at the Tg. The fractional value of the total volume is often taken as 0.025, which is so small that segmental jumps become impossible below the Tg.

The free volume (Vf) of the liquid is defined by Vf = VT – Vo, where VT is the total volume of the liquid at temperature T and V₀ is the theoretical molar volume or occupied volume. The total volume is the sum of the free volume (Vf) and of the occupied volume (V₀). The occupied volume (V₀) includes the van der Waals radii plus the fluctuation volume which is related to the thermal vibrational and rotational motion of the molecule. Thus, the Tg can be viewed as accessing by segmental jumps of the macromolecular chain segments into vacant spaces not occupied by the polymer. The higher the free volume the more easily the jumps can occur and the lower the viscosity or more increase of fluidity

The Temperature Coefficient of Expansion of Free Volume (αf) = 1/vg (dV/dT) is given, where vg is the specific volume at Tg and dV/dT is the observed volume change as a function of temperature. The glass transition temperature Tg, as explained by the free volume theory, is the temperature at which the free volume Vf reaches a constant value.

For linear polymers, the Tg value is an increasing function of the molar mass, such that Tg varies linearly with the reciprocal of the number average molecular weight (Mn). This dependance is a result of the contribution of chain-end segments in molecular motion. As the number of chain-ends increases, the free volume increases due to increasing molecular motion, and therefore the Tg temperature decreases. This relationship is expressed in the following Flory-Fox empirical equation (1):

[picture] | (1)

where

Tg∞ is the glass transition temperature of an infinite molecular weight

Mn is the number-average molecular weight as g/mol

K is a constant given by, with units of °C mol/g

[picture] | (2)

where

Vc = the free volume contributed by chain ends in cm³

ρ = polymer density in g/cc

NA = Avogadro number 6.023 x ²³ molecules/mole

α = thermal expansion coefficient per °C.

2.4. Summary of the Chemical Structure Factors affecting Tg

Molecular weight for linear homopolymers: an increase in molar mass (molecular weight) leads to a decrease in chain end concentrations resulting in a decrease of free volume at the end group region and thus an increase in the glass transition temperature Tg. Addition of diluents or plasticizers increase free volume thus decreases Tg

Molecular structure: an insertion of bulky or rigid inflexible side groups, such as a phenyl group, will increase the glass transition temperature Tg due to the decrease in mobility/flexibility. Whereas introducing flexible side chains like acetate group decreases Tg

Chemical cross-linking: an increase in cross linking density decreases mobility, leading to a decrease in free volume and thus an increase in the glass transition temperature Tg. Increasing branching increase free volume thus decrease Tg

Polar groups: the presence of polar groups increases the dipole-dipole intermolecular forces which increases interchain attraction and cohesion, leading to a decrease in free volume resulting in an increase in Tg.

Cohesive-Energy Density: Increase tacticity decreases Tg whereas increase in cohesive energy density increase Tg.

2.5. The Determination of Tg

Experimental methods of measuring the glass transition (Tg) are based on the physical properties of the polymers converted from the glassy state to the soft rubbery state. Three overall methods have been used:

1. Experiments defined by equilibrium thermodynamics or the Steady-State method, in which the physical properties are measured under static isothermal equilibrium conditions over a temperature range including Tg. Examples include dilatometry, penetrometry, refractometry, calorimetry-specific heat such as DSC or thermal analysis.

2. Experiments defined by dynamic or transport properties or the Dynamic Method, in which the physical polymer properties are measured during the heating of the polymer above Tg, in these methods Tg is measured by extrapolation to obtain isothermal conditions. Examples include: Infrared spectroscopy, NMR, Stress birefringence, dielectric loss, stress relaxation, Dynamic mechanical properties.

3. Tests related to end-use properties examples include impact resistance, softening point and hardness measurement.

2.6. Polystyrene Glass Transition

One polymer that has been previously studied extensively with respect to glass transition temperatures and physical and chemical properties is polystyrene [19,20]. Polystyrene is a solid white thermoplastic substance (material) existing as an atactic non-crystalline linear homopolymer that is in a glass-like state at room temperature. The glass formation is due to the lack of structural regularity in the polystyrene molecules. Below its glass transition temperature, polystyrene exhibits as a hard and stiff, yet brittle state and has frozen glass-like properties at room temperature. It has moderate to high dipole-dipole intermolecular forces and is a vitreous low mechanical strength material.

3. Materials and Methods

3.1. Overview: Overall Procedure

Five mg samples of linear atatic monodispersed homopolymers of polystyrene ranging from 3,250 – 950,000g/mol were prepared in Aluminum DSC pans. Additionally, mixtures of approximately 20%, 40%, 60%, and 80% by mass of either 3,250, 10,100 or 156,000 g/mol polystyrene were also prepared. Aluminum pans were weighed before and after each addition of polystyrene to the pan to determine weight % composition. Thermograms were obtained on both the individual and binary mixture samples to determine the glass transition temperatures. For the individual samples, a plot of the glass transition temperature as a function of molecular weight was obtained and the K constant in the Flory Fox equation was calculated. For the mixtures, correlation curves were developed between the experimentally and theoretically determined glass transition temperatures. The correlation curves were used to determine the proportions of two known polystyrene components based on the Tg of a binary sample created.

3.2. Experimental Materials

Monodispersed, linear, atatic, non-crystalline homopolymer samples of polystyrene of various molecular weights were purchased from Polymer Laboratories (PL) (later purchased by Agilent technologies) with polydispersity indexes less than 1.05. All the samples of polystyrene were synthesized using proprietary anionic polymerization techniques. PL polymer standards were all prepared under rigorous synthetic conditions and characterized by the most modern absolute and chromatographic techniques. The polymers are labelled with their PEAK molecular weights (Mp) as determined by high performance Gel Permeation Chromatography (HP-GPC) on PL gel column for Mp and Mw /Mn. All samples were used without purification. The aluminum pans used in this study had an internal volume of 20 microliters.

3.2.1. Differential Scanning Calorimetry

The Glass Transition temperature (Tg) results were obtained using a Perkin-Elmer power compensated Differential Scanning Calorimeter (DSC) model Pyris 1. The DSC was used in its high temperature mode. Calibration of the thermal outputs of the DSC were obtained using an empty reference Al pan. Prior to beginning the experiment, the DSC was calibrated for Temperature, heat flow and baseline linearity. This was done by first running empty cells in both the sample and reference compartments to produce a thermal baseline. Highly pure standards of tin, lead and indium were run thru 4 thermal cycles/ramps of two heating and two cooling at a constant rate of 10.0°C/min. The onset melting and recrystallization temperatures for the standards were used for temperature calibration. The onset melting/recrystallization temperature are defined as the temperature at the initial endothermic/exothermic change from the thermal baseline. The change in enthalpy (ΔH) was used to calibrate heat flow. The ΔH is found by the peak area under the curve. The endothermic and exothermic transition temperatures as well as the Enthalpies of Fusion and crystallization were recorded by the Pyris 1 for Windows software. Table 1 summarizes the properties of the polystyrene samples studied. Table 1 describes the batch #, polydispersity index and the peak molecular weight (g/mol) of the samples that were analyzed. All samples underwent two heating cycles and one cooling cycle between 50.0 – 120.0°C at a rate of 10°C/min for heating and cooling. The glass transition temperatures and enthalpies of transition were determined using the Perkin-Elmer thermal analysis software Pyris for Windows. The 2nd heating thermogram was used to determine the onset, mid-point-1/2 Cp and endpoint temperatures of Tg, as well as endothermic enthalpies of transition. The first heating cycle was used to erase the thermal history of the polystyrene samples. All experiments were run under dry nitrogen flowing at 20 cc/min. The flowing nitrogen was used to prevent any moisture pickup or oxidative degradation. The experimental analysis is not limited to this specific DSC hardware and software.

3.2.2. Experimental Procedure

Ten 5 mg samples of polystyrene ranging from 3,250 – 950,000 g/mol were studied as shown in Table 1. Sixteen 10 milligram binary mixtures of 3,250, 10,100 or 156,000 g/mol polystyrene samples were studied as shown in Table 2. Each sample was added to a 20-microliter aluminum pan using a metal spatula. All the pans were labeled with the gravimetric ratio of samples in the mixture. Following each addition, the mass of the pan was recorded using an analytical balance having a precision of 0.1 mg. The weight percent composition of the mixtures were determined using these masses. All mixtures were analyzed using DSC with an empty reference pan standard. Glass transition temperatures and enthalpies of fusion were recorded.

3.2.3. Hazards

Polystyrene (CAS# 9003-53-6) is regularly used as a versatile plastic in food packaging, containers and bottles. The FDA has approved polystyrene for use in contact with food, as it does not pose any significant health risks unless consumed at high levels. Goggles and gloves are nevertheless required to avoid exposure to the eyes and skin. Waste solutions should be disposed of according to EPA and local guidelines.

4. Results and Discussion

Table 1 summarizes the glass transition data. Table 2 details the onset and ½ Cp glass transition temperatures (Tg), and Enthalpy of transitions for each of the polystyrene samples studied.

Table 1. Summary of Polystyrene Samples

Table 2. Summary of Polystyrene Mixtures

Figure 2 shows the glass transition temperature as a function of molecular weight in logarithmic scale for the various polystyrene samples. The graph shows that as molecular weight increases, glass transition temperature also increases until a certain molecular weight, after which the Tg plateaus. Additionally, the rate at which the Tg is increasing decreases as molecular weight is increased. The Tg max for the samples was 112.11°C, which was obtained from the 950,000 g/mol polystyrene sample.

Figure 2. Glass transition temperature (Tg) vs molecular weight

Figure 3 is an overlay of the DSC thermograms for the 5500, 10,100, 66,000, and 330,000 g/mol polystyrene samples. As shown in the figure, the samples with a lower molecular weight had a lower Tg and a larger width for the range of the glass transition region. Furthermore, the lower molecular weight PS samples generally had higher endothermic heat flows than the larger polystyrene samples.

Figure 3. DSC heat trace of glass transition thermograms showing heat flow vs temperature

Figure 4 shows the glass transition temperature as a function of the reciprocal of molecular weight. Based on the equation Tg = Tg,∞ - K / Mn, the y-intercept of the graph is Tg,∞ and the slope is -K. The experimental value of K was found to be 1.083x105 °C*g/mol, which aligns very closely with the experimental value of K as determined by Flory and Fox to be 1.000x105 °C*g/mol (5,6). Likewise, the Tg,∞ experimental of the line of best fit is 109.22°C, which is very close to the theoretical value of 108°C.

Figure 4. Glass transition temperature vs reciprocal molecular weight

Figure 5 – Figure 7 show the experimental and theoretical glass transition temperatures for the binary mixtures of 3250, 10100, 156000 g/mol polystyrenes. Theoretical Tgs were calculated using the Fox equation which relates

[picture] | (3)

where wt. refers to the weight fraction of the polystyrene mixtures. In mixtures, a diluent of lower molecular weight increases the free volume of the sample which decreases Tg. For all mixtures, the experimental and theoretical curves aligned very closely with each other having a correlation coefficient of .98 or greater.

Figure 5. Glass transition temperature of 3,250 and 10,100 g/mol binary mixture vs weight percent of 10,100 g/mol polystyrene. The y-intercept is the glass transition of the lower weight diluent, 3250 g/mol polystyrene

Figure 6. Glass transition temperature of 3,250 and 156,000 g/mol binary mixture vs weight percent of 156,000 g/mol polystyrene. The y-intercept is the glass transition of the lower weight diluent, 3250 g/mol polystyrene.

Figure 7. Glass transition temperature of 10,100 and 156,000 g/mol binary mixture vs weight percent of 156,000 g/mol polystyrene. The y-intercept is the glass transition of the lower weight diluent, 10,100 g/mol polystyrene

Figure 8. Enthalpy of transition vs molecular weight

5. Conclusions

The experiment corroborates the Flory-Fox equation which relates molecular weight to glass transition temperature. As molecular weight increased, glass transition temperature increased until a certain molecular weight after which the glass transition temperature leveled off. There is a strong, linear correlation between the gravimetric compositions and the glass transition temperatures found using DSC. The experiment serves as an excellent tool for the undergraduate polymer chemistry laboratory as the methodology can be readily adopted for similar experiments with different polymers.

Acknowledgments

We acknowledge the support from a Hofstra University HCLAS Faculty Research and Development Grant.

List of Abbreviations and Symbols

DSC – Differential Scanning Calorimetry

Tg – Glass Transition Temperature

PS – Polystyrene

PDI – Polydispersity Index

Wt1 0r 2 – weight fraction

Tg∞ - Glass Transition Temperature at Infinite Molecular Weight

K -Flory-Fox Constant (°Cg/mol)

Mn -Number Average Molecular Weight

Vf - Free Volume

ρ - density

Na – Avogadro’s number

α – coefficient of Thermal Expansion

mW – milli watts

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