^{1,a)}, Jack F. Douglas

^{2}and Karl F. Freed

^{1}

### Abstract

The entropy theory of glass formation, which has previously been developed to describe general classes of polymeric glass-forming liquids, is extended here to model the thermodynamic and dynamic properties of poly(-olefins). By combining this thermodynamic theory with the Adam–Gibbs model (which relates the configurational entropy to the rate of structural relaxation), we provide systematic computations for all four characteristic temperatures (, , , ), governing the position and breadth of the glass transition, and the fragility parameters describing the strength of the temperature dependence of the structuralrelaxation time, where is the temperature below which the relaxation is non-Arrhenius, is the crossover or empirical mode-coupling temperature, is the glass transition temperature, and is the temperature at which the extrapolated relaxation time diverges. These temperatures and fragility parameters are evaluated as a function of molar mass, pressure, and the length of the -olefin side chains. The nearest neighbor interaction energy and local chain rigidities are found to strongly influence the four characteristic temperatures and the low temperature fragility. We also observe an “internal plasticization” of the poly(-olefins) wherein the fragility decreases as the number of “flexible” side group units increases. Our computations provide solid support for a pressure counterpart of the Vogel–Fulcher–Tammann relation. The entropy theory of glass formation predicts systematic changes in fragility with chain stiffness, cohesive energy, polymerization index, and side chain length, and qualitative trends in these parameters are discussed.

This research is supported in part by NSF (Grant No. CHE-0749788), PRF (Grant No. 46666-AC7), and by the Joint Theory Institute, which is funded by the University of Chicago and Argonne National Laboratory. We are grateful to Jacek Dudowicz for several helpful discussions. This is an official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States.

I. GLASS TRANSITION PROBED BY ENTROPY THEORY

II. INFLUENCE OF BENDING AND COHESIVE ENERGIES ON GLASS FORMATION

III. TWO-BENDING ENERGY MODEL FOR POLY(-OLEFINS): DESCRIPTION OF MODELS AND CHOICE OF PARAMETERS

IV. GLASS TRANSITION TEMPERATURES AND FRAGILITIES: INFLUENCE OF MOLECULAR STRUCTURE

V. INFLUENCE OF MOLAR MASS ON GLASS FORMATION AT AMBIENT PRESSURE

VI. INFLUENCE OF HYDROSTATIC PRESSURE ON GLASS FORMATION

A. Pressure dependence of glass transition temperatures

B. Pressure dependence of the fragilities

VII. PRESSURE ANALOG OF THE VOGEL–FULCHER–TAMMANN EQUATION

VIII. CONCLUSIONS

### Key Topics

- Polymers
- 56.0
- Glass transitions
- 44.0
- Entropy
- 25.0
- Activation energies
- 14.0
- Relaxation times
- 13.0

## Figures

Temperature dependence of activation energy, reduced by its high temperature limiting value over a broad range of temperatures computed for a polymer melt of PP chains using the entropy theory for ambient pressure (solid line). Dotted and dashed lines represent the low (VFT) and high-temperature non-Arrhenius regimes of structural relaxation, respectively. The analytical approximation for computed using the polynomial coefficients (see text) superimposes within the thickness of the solid line and is not presented. An arrow marks the onset temperature above which demonstrates an Arrhenius behavior. The inset displays the temperature dependence of for PP melt in the form of the Angell plot.

Temperature dependence of activation energy, reduced by its high temperature limiting value over a broad range of temperatures computed for a polymer melt of PP chains using the entropy theory for ambient pressure (solid line). Dotted and dashed lines represent the low (VFT) and high-temperature non-Arrhenius regimes of structural relaxation, respectively. The analytical approximation for computed using the polynomial coefficients (see text) superimposes within the thickness of the solid line and is not presented. An arrow marks the onset temperature above which demonstrates an Arrhenius behavior. The inset displays the temperature dependence of for PP melt in the form of the Angell plot.

Influence of backbone bending energy on fragility parameter and glass transition temperature for the fixed cohesive energy , chain length , pressure (; ), and molecular structure (PP).

Influence of backbone bending energy on fragility parameter and glass transition temperature for the fixed cohesive energy , chain length , pressure (; ), and molecular structure (PP).

Influence of cohesive energy on fragility parameter and glass transition temperature for fixed bending energy , chain length , pressure , and molecular structure (PP).

Influence of cohesive energy on fragility parameter and glass transition temperature for fixed bending energy , chain length , pressure , and molecular structure (PP).

Correlation of fragility parameters and fragility (in part a) and glass transition temperature (in part b) with the fraction of excess free volume evaluated at the respective . Only one parameter varies for each curve: the bending energy , the cohesive energy , the chain length , or the pressure . The molecular structure (PP) is fixed.

Correlation of fragility parameters and fragility (in part a) and glass transition temperature (in part b) with the fraction of excess free volume evaluated at the respective . Only one parameter varies for each curve: the bending energy , the cohesive energy , the chain length , or the pressure . The molecular structure (PP) is fixed.

(a) Correlation between the fragility and glass transition temperature at fixed cohesive energy , pressure , and molecular structure (PP). The squares compile the data from Fig. 2 for and for variable , while the other two curves provide for (diamonds) and (circles) for variable. The solid lines in all cases are linear fits with and , and , and and . (b) The correlation between the fragility and with the variation in cohesive energy (circles) at fixed backbone bending energy , pressure , and molecular structure (PP). The solid line is a linear fit with and .

(a) Correlation between the fragility and glass transition temperature at fixed cohesive energy , pressure , and molecular structure (PP). The squares compile the data from Fig. 2 for and for variable , while the other two curves provide for (diamonds) and (circles) for variable. The solid lines in all cases are linear fits with and , and , and and . (b) The correlation between the fragility and with the variation in cohesive energy (circles) at fixed backbone bending energy , pressure , and molecular structure (PP). The solid line is a linear fit with and .

Fragility of polymer melts as a function of the bending energy ratio. The reference F-F model is defined by . The F-S and S-F models ascribe the bending energy of the more rigid subchain (either backbone or side group) as variable, while the stiffness of the flexible subchain is fixed as . The bottom inset depicts the monomer topology for the S-F, F-S, and F-F classes of polymers.

Fragility of polymer melts as a function of the bending energy ratio. The reference F-F model is defined by . The F-S and S-F models ascribe the bending energy of the more rigid subchain (either backbone or side group) as variable, while the stiffness of the flexible subchain is fixed as . The bottom inset depicts the monomer topology for the S-F, F-S, and F-F classes of polymers.

Specific molecular volumes of high molecular mass PP as a function of temperature computed with models I (dashed line) and II (dotted line) with experimental data for high molecular mass amorphous PP polymer (solid line). The part of the solid curve below 295 K is obtained by extrapolation of the analytical fit to the experimentally measured to low temperatures. See original Ref. 22 for discussion of experimental uncertainties.

Specific molecular volumes of high molecular mass PP as a function of temperature computed with models I (dashed line) and II (dotted line) with experimental data for high molecular mass amorphous PP polymer (solid line). The part of the solid curve below 295 K is obtained by extrapolation of the analytical fit to the experimentally measured to low temperatures. See original Ref. 22 for discussion of experimental uncertainties.

Glass transition temperature for poly(-olefins) as a function of the number of carbon atoms in the side groups for constant pressure and fixed chain lengths . Squares depict the computations within model I, diamonds correspond to model II, and circles are experimental data for PE, , and for PP, , through polynonene-1, (see text). See original Ref. 29 for discussion of experimental uncertainties.

Glass transition temperature for poly(-olefins) as a function of the number of carbon atoms in the side groups for constant pressure and fixed chain lengths . Squares depict the computations within model I, diamonds correspond to model II, and circles are experimental data for PE, , and for PP, , through polynonene-1, (see text). See original Ref. 29 for discussion of experimental uncertainties.

Fragility parameter for poly(-olefins) as a function of the number of carbon atoms in the side groups for constant pressure and fixed chain lengths . The computations are presented for model I (squares) and model II (diamonds), and the circles are experimental data for PE and PP. See original Refs. 61–63 and 17 for discussion of experimental uncertainties.

Fragility parameter for poly(-olefins) as a function of the number of carbon atoms in the side groups for constant pressure and fixed chain lengths . The computations are presented for model I (squares) and model II (diamonds), and the circles are experimental data for PE and PP. See original Refs. 61–63 and 17 for discussion of experimental uncertainties.

Molar mass dependence of the glass transition temperature for PP at fixed pressure . Squares and diamonds correspond to models I and II, respectively. For comparison, experimental data for of PS are displayed as empty circles. The temperatures are normalized by their corresponding high molar mass limits . The inset depicts the slopes of this linear scaling recomputed for the dependence of the inverse glass transition temperatures on the inverse polymerization indices in Eq. (5) for several poly(-olefins) as a function of the number of carbon atoms in the side groups . The inset compiles the computations within model I (filled circles). The dotted line in the inset depicts the corresponding slope for PS obtained from experimental data for . See original Ref. 32 for discussion of experimental uncertainties for PS.

Molar mass dependence of the glass transition temperature for PP at fixed pressure . Squares and diamonds correspond to models I and II, respectively. For comparison, experimental data for of PS are displayed as empty circles. The temperatures are normalized by their corresponding high molar mass limits . The inset depicts the slopes of this linear scaling recomputed for the dependence of the inverse glass transition temperatures on the inverse polymerization indices in Eq. (5) for several poly(-olefins) as a function of the number of carbon atoms in the side groups . The inset compiles the computations within model I (filled circles). The dotted line in the inset depicts the corresponding slope for PS obtained from experimental data for . See original Ref. 32 for discussion of experimental uncertainties for PS.

Molar mass dependence of the fragility parameter for PP at fixed pressure from model I (squares) and model II (diamonds). The fragility parameters are normalized by their corresponding high molar mass limits for each model. The inset presents the chain length dependence of the computed steepness indices for PP at fixed pressure and models I (squares) and II (diamonds) along with the experimental fragility of PS chains as functions of the inverse polymerization index for each polymer (PP or PS). See original Ref. 32 for discussion of experimental uncertainties for PS.

Molar mass dependence of the fragility parameter for PP at fixed pressure from model I (squares) and model II (diamonds). The fragility parameters are normalized by their corresponding high molar mass limits for each model. The inset presents the chain length dependence of the computed steepness indices for PP at fixed pressure and models I (squares) and II (diamonds) along with the experimental fragility of PS chains as functions of the inverse polymerization index for each polymer (PP or PS). See original Ref. 32 for discussion of experimental uncertainties for PS.

Pressure dependences of the computed Kauzmann temperature (squares), Vogel temperature (diamonds), and the corresponding pressure parameter (filled triangles). The pressure dependence of (circles) is computed using the conventional definition of the glass transition temperature . The computations correspond to high molecular mass PP chains and model I (part a) and model II (part b). Experimental data from Ref. 40 are presented by empty triangles. The errors of these data are smaller than the size of the symbols.

Pressure dependences of the computed Kauzmann temperature (squares), Vogel temperature (diamonds), and the corresponding pressure parameter (filled triangles). The pressure dependence of (circles) is computed using the conventional definition of the glass transition temperature . The computations correspond to high molecular mass PP chains and model I (part a) and model II (part b). Experimental data from Ref. 40 are presented by empty triangles. The errors of these data are smaller than the size of the symbols.

Fragility parameter for high molar mass PP chains as a function of pressure computed within models I (squares) and II (diamonds). The inset depicts the pressure dependence of the steepness index for the same system computed from using Eq. (10). The legends marking the models are identical to those used in the main part of illustration.

Fragility parameter for high molar mass PP chains as a function of pressure computed within models I (squares) and II (diamonds). The inset depicts the pressure dependence of the steepness index for the same system computed from using Eq. (10). The legends marking the models are identical to those used in the main part of illustration.

Linearization of Eq. (15) in the pressure domain which is used for the evaluation of the parameters, the ideal glass transition pressure and the isothermal pressure fragility parameter , for the PVFT equation. Each curve (isotherm) corresponds to the specific temperature presented in the legend. The symbol in each series is as a function of pressure computed for PP polymer within model I. The lines are the linear fits to .

Linearization of Eq. (15) in the pressure domain which is used for the evaluation of the parameters, the ideal glass transition pressure and the isothermal pressure fragility parameter , for the PVFT equation. Each curve (isotherm) corresponds to the specific temperature presented in the legend. The symbol in each series is as a function of pressure computed for PP polymer within model I. The lines are the linear fits to .

Structural relaxation times for a PP polymer melt computed as a function of pressure at fixed temperatures for model I (isotherms). The LCT computations (symbols) are presented along with their analytical fits using the PVFT equation (lines). The isotherms are marked in the legend.

Structural relaxation times for a PP polymer melt computed as a function of pressure at fixed temperatures for model I (isotherms). The LCT computations (symbols) are presented along with their analytical fits using the PVFT equation (lines). The isotherms are marked in the legend.

## Tables

Glass formation properties for some poly(-olefins) .

Glass formation properties for some poly(-olefins) .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content