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Retrieved 31 March Reviews of Modern Physics. Bibcode : RvMPS.. Princeton University. November World Scientific Newsletter. Next article. Related articles. Share this page. Glassy substances are everywhere, yet this state of matter poses many puzzles. The basic picture is clear enough -- glasses are solids that lack the regular atomic structure of a crystal.

How and why they form, however, are questions that have kept physicists busy for decades. Now, research from Japan has shown that glass formation can be understood if liquid structure is properly described. In a well-designed set of molecular dynamics simulations, they aimed to resolve whether the process is fundamentally thermodynamic dependent on some form of static order or dynamic driven by random atomic motions. The team simulated supercooled liquids near the transition point, the temperature where particle diffusion stops and an amorphous solid appears.

The aim was to find a link between structural patterns and the slowing of atomic motion, i. If it exists, this structure-dynamics correlation would verify that thermodynamics controls the formation of glasses, just as for crystals. That would be a major step toward a universal theory. However, since glasses are apparently long-range disordered, the importance of local order is elusive. In each simulation, the team quantified how well the atoms packed together in the cooling liquid by measuring a structural order parameter.

Book file PDF easily for everyone and every device. This Book have some digital formats such us :paperbook, ebook, kindle, epub, fb2 and another formats. Please note: Thus, the changes of a physical system as viewed at different size scales can be investigated systematically. Thermodynamic picture of the glassy state Cold atoms in optical lattices are used as quantum simulators , that is, they act as controllable systems that can model behavior of more complicated systems, such as frustrated magnets.

Submission history Retrieved 20 April Physics in Perspective. Bibcode : PhP References Related articles 0.

## Model of the glass transition and the glassy state | SpringerLink

Figures 0. Particularly, the change in glass transition temperature as a function of pressure and its connection with the thermodynamic coefficients, which provides a suitable parameter to elucidate the nature of glass transition, has been intensively examined. Numerous experimental studies on different glass-formers reveal that the pressure coefficient of the glass transition temperature is substantial in the case van der Waals liquids i. Certainly, the development of a suitable universal relation between and the key thermodynamic coefficients i.

In the past, many efforts have been gained to explain the nature of the glass transition. It is well established, that at the glass transition, Gibbs free energy and its first derivatives i. Eventually, attempts have been made to consider the liquid-glass transition as a second-order phase transition. Some efforts were undertaken to verify both Ehrenfest equations eqs. However, it is worth noting that in the case of a second-order phase transition the system goes from one equilibrium state to the other, whereas at the glass transition, the system is transferred from the metastable equilibrium into the non-equilibrium state, which is the glass.

Another approach to describe the liquid-glass transition, which is certainly worthy of attention, is the concept of order parameters, which was introduced by Donder and van Rysselberghe It suggests that the state of the system in both the equilibrium and non-equilibrium states depends on the thermodynamic intensive parameters temperature, pressure and a number of order parameters.

Then, Simon proposed that the glass transition could be considered as a process at which the system becomes suddenly kinetically frozen in 12 , i. Assuming that only one order parameter is sufficient to describe the structural differences between the liquid and the glass, both Ehrenfest equations and hence value were theoretically obtained by Davies and Jones 5 , 10 , 13 , In this context, it has to be noted that equation 1 , which is usually not fulfilled 14 , 15 , 16 , 17 , 18 , incorporates also measured in the nonequilibrium glassy state, which makes it difficult to verify experimentally On the other hand, equation 2 seems to hold reasonably well for many systems, although not for all 20 , Nevertheless, the Prigogine-Defay ratio is seen as an indicator of the complexity; the number suggesting the degree to which the liquid fails to be described by a single order parameter This behavior occurs when the time scale of changes of the external parameter become comparable with the characteristic relaxation time of the system e.

Therefore, the transformation from the liquid to the glassy state is expected to have a kinetic, rather than thermodynamic, origin. It is consistent with the one of the most important conclusions from the long history of the research on the glass transition process, i. Hence, many efforts have been made to formulate the definition of the glass transition by considering it as a purely kinetic process. First attempts were made by Bartenev 24 and Jones 25 , in Two years later, from a general examination of the cooling process 24 , 26 , Bartenev derived the relation 24 , which was experimentally corroborated for different materials 27 , It is worth noting that an identical formula can be obtained from the chemical reaction model when one employs certain additional conditions, which are commonly met in the glass transition, as Volkenstein and Ptizyn presented Nevertheless, later and detailed examinations performed by Moynihan et al.

Thereby, the general understanding has been shaped, according to which, only the value of defines the glass transition, if the constant cooling rate is applied to different isobaric states. In this context, the interesting considerations given by Hodge 32 , to the Deborah number, which is defined as the ratio of timescales of the observed and the observer, , are worthy of mentioning. According to them the glass transition is seen when the above two timescales cross over and then Deborah number passes through unity. Then in the temperature domain, that is explored most thoroughly, a Deborah number of unity that defines an average can be expressed in terms of the rate of change of some characteristic timescale , determined during cooling:.

Since the temperature dependence of relaxation time for many processes is given by the empirical Vogel-Tamman-Fulcher equation, at the isobaric conditions the above criterion results in the following relation , where is the isobaric fragility.

Assuming that the glass transition is approached at a constant cooling rate, becomes proportional to. Then, combining it with Bartenev relation, one obtains that the constant value depends on , which implies that values of do not have to be invariant for all isobaric conditions. Hence, the glass transition can be defined, with very good agreement, by a constant value of which for simplicity in the later part will be denoted by.

## Mechanics of Glassy Polymers

Following the general wisdom, in this paper we examine the glass transition as a kinetic process and consequently on this basis, we derive one universal equation for the pressure coefficient of the glass transition temperature. We show that by considering the dependence of the characteristic relaxation of the system, defined by the structural relaxation time , on different external parameters, one can obtain equations for , which originates from one universal equation.

In this context, they are not independent, and hence the relation equivalent to the Prigogine-Defay ratio cannot be established for the glass transition, at least without employment of an additional condition met in this process. The three above possible equations for , obtained in the cases of dependences, , , and , have been verified. We also show that predictions based on the above equations are consistent within a wide range of pressures. The structural relaxation time depends on many thermodynamic quantities, e. However, only two of them can be changed independently from the others.

Since temperature is the physical quantity most often controlled in different experiments, we propose that in the most convenient way, the structural relaxation time can be considered as a function of and another thermodynamic quantity. Then, the complete differential of the structural relaxation time , at the glass transition defined at a constant value of , equals.

On the other hand, is a function of two independent thermodynamic quantities that can be selected from many of the other thermodynamic quantities. One of the most natural selections is and , and then. Using the above relation between thermodynamic quantities we can rewrite the complete derivative of and then. Since, , the latter equation transforms to , where expressions for the isobaric and the isothermal fragility can be exploited.

As we have mentioned above, is a function of and , thus the formula for its complete derivative is expressed as follows. Using it, we obtain the required equation for the pressure coefficient of the glass transition temperature, which takes the subsequent form.

The term, which appears in the denominator and consists of quotient of the isobaric and the isothermal fragility, can be transformed to , where is the isochoric fragility. In conclusion, the general equation for the pressure coefficient of the glass transition temperature takes the following form. Thus, one can easily obtain several expressions for , e. The predictions of above equations for and values of this coefficient obtained from analysis of the experimental measurements are presented in Fig. In this context, it has to be noted that equation 5 was earlier successfully verified at ambient pressure for many glass-forming liquids that belong to the different material groups Values of the pressure coefficient of the glass transition temperature calculated from equations 5 , 6 , 7 , respectively, and obtained from analysis of experimental measurements exp.

It is worth noting that the universal character of the equivalent equations 5 — 7 , can be extended beyond the assumed case of the dependence.

### Acknowledgements

Considering the density scaling laws for and discussed later for and in the Supplementary Information for , we have very recently argued 34 that the structural relaxation time can be in general a volume-entropy function. Then, equation 5 also remains valid as shown in Supplementary Information.

The excellent agreement between values of predicted by equations 5 — 7 , is not surprise since all equations are received from equation 4. It has to be mentioned that employing the relation between fragilities 35 ; one can easily transform equation 6 to equation 5 as well as by using the thermodynamic relation , equation 7 can be converted to equation 5.

## Glass Transition Thermodynamics and Kinetics

Taking the above into account, we can expect that knowledge of the relationship between any and enables transformation of whichever equation for resulted from equation 4 to equation 5. Besides, derivation of two independent equations for , and an establishment of the relation among the thermodynamic coefficients at the glass transition corresponding to the Prigogine-Defay ratio , is not possible, at least based on only the kinetic description of this process.

So, in order to establish the relation equivalent to equation 3 that is satisfied at the glass transition, we must employ an additional condition met in this process. This has been done in Ref.

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In the next paragraph we present the derivation of the Ehrenfest equations from our universal equation for the pressure coefficient of the glass transition temperature. However, now we can mention that this procedure requires additional assumptions and hence equation 5 from ref. Therefore, the main equation in the mentioned paper, i. Then, and of the metastable liquid and the glass have the same values, which implies that the configurational volume and entropy equal 0.

Since, introduced by us in equation 4 , can be any thermodynamic quantity, there is nothing to preclude the consideration dependence of the characteristic relaxation time of the system on configurational values e. In this way, equation 4 takes following forms. Now, it can be seen that equations 8 and 9 will become identical with equations 1 and 2 if the quotient of isochoric and isothermal fragility equals 0, which implies that or.

Both the restrictions reflect one possible but limited behavior of the molecular dynamics, i. Moreover, it has to be noted that the Ehrenfest equations are limiting cases of our universal equation for , which seems to be much more general. On the other hand, if only fluctuations of the temperature govern the molecular dynamics, and behave conversely to before, and hence , which indeed reflects the situation at which the glass transition occurs at constant temperature independently of pressure and hence volume.