^{1}, Jacek Dudowicz

^{2}and Karl F. Freed

^{2,a)}

### Abstract

A significant body of evidence indicates that particles with excessively high or low mobility relative to Brownian particles form in dynamic equilibrium in glass-forming liquids. We examine whether these “dynamic heterogeneities” can be identified with a kind of *equilibrium polymerization*. This correspondence is first checked by demonstrating the presence of a striking resemblance between the temperature dependences of the configurational entropy in both the theory of equilibrium polymerization and the generalized entropytheory of glass formation in polymer melts. Moreover, the multiple characteristic temperatures of glass formation are also shown to have analogs in the thermodynamics of equilibrium polymerization, supporting the contention that both processes are varieties of rounded thermodynamictransitions. We also find that the average cluster mass (or degree of polymerization) varies in nearly inverse proportionality to . This inverse relation accords with the basic hypothesis of Adam-Gibbs that the number of particles in the cooperatively rearranging regions (CRR) of glass-forming liquids scales inversely to of the fluid. Our identification of the CRR with equilibrium polymers is further supported by simulations for a variety of glass-forming liquids that verify the existence of stringlike or polymeric clusters exhibiting collective particle motion. Moreover, these dynamical clusters have an exponential length distribution, and the average “string” length grows upon cooling according to the predictions of equilibrium polymerizationtheory. The observed scale of dynamic heterogeneity in glass-forming liquids is found to be consistent with this type of self-assembly process. Both experiments and simulations have revealed remarkable similarities between the dynamical properties of self-assembling and glass-forming liquids, suggesting that the development of a theory for the dynamics of self-assembling fluids will also enhance our understanding of relaxation in glass-forming liquids.

This work is supported in part by NSF Grant No. CHE 0416017.

I. INTRODUCTION

II. FLORY-HUGGINS THEORY OF EQUILIBRIUM POLYMERIZATION

A. Free association model

B. Activated equilibrium polymerization

III. GLASS FORMATION AND GENERALIZED ENTROPYTHEORY OF POLYMERGLASS FORMATION

A. Basic components of the generalized entropytheory of polymerglass formation

B. Characteristic temperatures of glass formation

IV. STRINGS, DYNAMICAL COOPERATIVELY REARRAGING REGIONS, AND EQUILIBRIUM POLYMERIZATION

A. Configurational entropy and dynamic cluster size distribution in equilibrium polymerization and glass-forming liquids

B. Characteristic temperatures of polymerization and glass formation

C. The analog of “fragility” in equilibrium polymerization

D. Comparison of size distribution of equilibrium polymers and strings of cooperative motion

E. Saturation of the self-assembly process at low temperatures and its implications

F. Two-state models of glass formation and self-assembly

G. Size of cooperatively rearranging regions and the average degree of polymerization: Evidence for universality at the crossover temperature

H. Equilibrium polymerization of immobile particle clusters

I. Equilibrium polymerization and Fisher clusters in glass-forming liquids

J. Implications of correspondence between equilibrium polymerization and glass formation for understanding the dynamics of associating fluids

V. CONCLUSIONS

### Key Topics

- Polymerization
- 152.0
- Polymers
- 102.0
- Glass transitions
- 65.0
- Entropy
- 56.0
- Polymer structure
- 30.0

## Figures

Temperature dependence of the reduced configurational entropy per lattice site for an incompressible equilibrium polymerization solution. for the FA and A models of equilibrium polymerization is calculated from Eqs. (4) and (12), respectively, and this quantity is shown in the reduced form specified by Eq. (13). These illustrative calculations (as well as those in Figs. 2–6) are performed for completely flexible chains. Curves 1 and 2 refer to the FA model with the enthalpy and entropy of polymerization chosen as and , and and , respectively. The third curve corresponds to the activated equilibrium polymerization model that is specified by , , and (with and being the enthalpy and the entropy of activation, respectively). The above free energy estimates are consistent with previous studies of equilibrium polymerization (Refs. 33, 41, 42, and 73), and the initial monomer concentration is taken as for specific comparison to string formation in glass-forming liquids (see text). Solid symbols denote the calculated saturation temperature , the inflection point temperature where exhibits an inflection point, and the crossover temperature . All temperatures are defined in the text. Open circles designate the polymerization transition temperatures estimated from the maximum of the specific heat .

Temperature dependence of the reduced configurational entropy per lattice site for an incompressible equilibrium polymerization solution. for the FA and A models of equilibrium polymerization is calculated from Eqs. (4) and (12), respectively, and this quantity is shown in the reduced form specified by Eq. (13). These illustrative calculations (as well as those in Figs. 2–6) are performed for completely flexible chains. Curves 1 and 2 refer to the FA model with the enthalpy and entropy of polymerization chosen as and , and and , respectively. The third curve corresponds to the activated equilibrium polymerization model that is specified by , , and (with and being the enthalpy and the entropy of activation, respectively). The above free energy estimates are consistent with previous studies of equilibrium polymerization (Refs. 33, 41, 42, and 73), and the initial monomer concentration is taken as for specific comparison to string formation in glass-forming liquids (see text). Solid symbols denote the calculated saturation temperature , the inflection point temperature where exhibits an inflection point, and the crossover temperature . All temperatures are defined in the text. Open circles designate the polymerization transition temperatures estimated from the maximum of the specific heat .

The configurational entropy calculated from the generalized entropy of glass formation as a function of temperature for representative high molar mass F–F and F–S polymer fluids at a constant pressure of (Ref. 48). The configurational entropy is normalized by its maximum value and is determined from Eq. (20) of Ref. 28. The characteristic temperatures of glass formation, , , , and , are indicated in the figure. The dotted line schematically depicts a correction to the mean field configurational entropy. As noted by Wolfgardt *et al.* (Ref. 62), does not vanish, but instead achieves a small plateau value at low temperatures. An individual monomer of the F–F and F–S polymers contains two backbone segments and one side group with three units (like the united atom representation of 1-pentene) (Refs. 48 and 50). The F–F and F–S polymers represent chains with a flexible chain backbone and flexible side groups, and flexible chain and stiff side branches, respectively, as described in our schematic model of glass formation in polymer melts (Refs. 48–50). The bending energies and and the van der Waals interaction energy are chosen as (F–F polymers), , and (F–S polymers), while the van der Waals energy is selected to be common for these two polymer classes as . The volume associated with a single lattice site and the lattice coordination number are taken as and , respectively. Each backbone and side chain semiflexible bond pair is further taken to have of one *trans* and two *gauche* configurations.

The configurational entropy calculated from the generalized entropy of glass formation as a function of temperature for representative high molar mass F–F and F–S polymer fluids at a constant pressure of (Ref. 48). The configurational entropy is normalized by its maximum value and is determined from Eq. (20) of Ref. 28. The characteristic temperatures of glass formation, , , , and , are indicated in the figure. The dotted line schematically depicts a correction to the mean field configurational entropy. As noted by Wolfgardt *et al.* (Ref. 62), does not vanish, but instead achieves a small plateau value at low temperatures. An individual monomer of the F–F and F–S polymers contains two backbone segments and one side group with three units (like the united atom representation of 1-pentene) (Refs. 48 and 50). The F–F and F–S polymers represent chains with a flexible chain backbone and flexible side groups, and flexible chain and stiff side branches, respectively, as described in our schematic model of glass formation in polymer melts (Refs. 48–50). The bending energies and and the van der Waals interaction energy are chosen as (F–F polymers), , and (F–S polymers), while the van der Waals energy is selected to be common for these two polymer classes as . The volume associated with a single lattice site and the lattice coordination number are taken as and , respectively. Each backbone and side chain semiflexible bond pair is further taken to have of one *trans* and two *gauche* configurations.

The reduced configurational entropy of Eq. (13) as a function of the reciprocal of the average degree of polymerization for an incompressible solution of associating species undergoing equilibrium polymerization. The free energy parameters are the same as those employed in Fig. 1 (curve 1), while the enthalpy and entropy for monomer activation in the low probability activation model are chosen as and (Ref. 36). The solid line presents corresponding to the AG hypothesis of an inverse proportionality between the configurational entropy and the size of the cooperatively rearranging regions (CRR) in glass-forming liquids.

The reduced configurational entropy of Eq. (13) as a function of the reciprocal of the average degree of polymerization for an incompressible solution of associating species undergoing equilibrium polymerization. The free energy parameters are the same as those employed in Fig. 1 (curve 1), while the enthalpy and entropy for monomer activation in the low probability activation model are chosen as and (Ref. 36). The solid line presents corresponding to the AG hypothesis of an inverse proportionality between the configurational entropy and the size of the cooperatively rearranging regions (CRR) in glass-forming liquids.

(a) The reduced configurational entropy (solid line) and , where is the extent of polymerization, (dashed line) as functions of temperature for the FA equilibrium polymerization model. Solid symbols indicate the calculated saturation temperature , the polymerization temperature , and the crossover temperature . The free energy parameters and the initial monomer concentration are the same as in Fig. 1 (curve 1). (b) The reduced configurational entropy (solid line) and , where is the extent of polymerization, (dashed line) as functions of temperature for the activated equilibrium polymerization model. Solid symbols indicate the calculated saturation temperature , the polymerization temperature , and the crossover temperature . The free energy parameters and the initial monomer concentration are the same as in Fig. 1 (curve 3).

(a) The reduced configurational entropy (solid line) and , where is the extent of polymerization, (dashed line) as functions of temperature for the FA equilibrium polymerization model. Solid symbols indicate the calculated saturation temperature , the polymerization temperature , and the crossover temperature . The free energy parameters and the initial monomer concentration are the same as in Fig. 1 (curve 1). (b) The reduced configurational entropy (solid line) and , where is the extent of polymerization, (dashed line) as functions of temperature for the activated equilibrium polymerization model. Solid symbols indicate the calculated saturation temperature , the polymerization temperature , and the crossover temperature . The free energy parameters and the initial monomer concentration are the same as in Fig. 1 (curve 3).

Comparison of the size distribution of strings from simulations of supercooled liquids and from calculations for equilibrium linear polymers in the free association (FA) model [see Eqs. (14) and (15)]. Symbols denote the simulation data of Donati *et al.* (Ref. 3) for three different reduced temperatures , 0.480, and 0.451, while lines are the fits to the simulation data obtained from the FH equilibrium polymerization theory. The initial monomer concentration is taken as the maximum concentration (5%) of mobile Lennard-Jones particles in the simulated glass-forming liquid. The fitted values of the enthalpy and entropy of polymerization are and , respectively. The inset presents a comparison of simulation data (Ref. 3) (triangles) for the logarithm of the average string length vs the inverse reduced temperature . The predictions (solid line) of the equilibrium polymerization theory are generated for the same free energy parameters and initial monomer concentration as used in the fits shown in the main figure.

Comparison of the size distribution of strings from simulations of supercooled liquids and from calculations for equilibrium linear polymers in the free association (FA) model [see Eqs. (14) and (15)]. Symbols denote the simulation data of Donati *et al.* (Ref. 3) for three different reduced temperatures , 0.480, and 0.451, while lines are the fits to the simulation data obtained from the FH equilibrium polymerization theory. The initial monomer concentration is taken as the maximum concentration (5%) of mobile Lennard-Jones particles in the simulated glass-forming liquid. The fitted values of the enthalpy and entropy of polymerization are and , respectively. The inset presents a comparison of simulation data (Ref. 3) (triangles) for the logarithm of the average string length vs the inverse reduced temperature . The predictions (solid line) of the equilibrium polymerization theory are generated for the same free energy parameters and initial monomer concentration as used in the fits shown in the main figure.

Temperature variation of the average degree of polymerization for the model of activated equilibrium polymerization that is illustrated in Fig. 1 (see curve 3). The inset presents the reciprocal of as a function of the reduced configurational entropy of Eq. (13).

Temperature variation of the average degree of polymerization for the model of activated equilibrium polymerization that is illustrated in Fig. 1 (see curve 3). The inset presents the reciprocal of as a function of the reduced configurational entropy of Eq. (13).

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