Correlating Free-Volume Hole Distribution to the Glass Transition

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Correlating Free-Volume Hole Distribution to the Glass Transition Temperature of Epoxy Polymers Amin Aramoon, Timothy D. Breitzman, Christopher F Woodward, and Jaafar A. El-Awady J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b04147 • Publication Date (Web): 16 Aug 2017 Downloaded from http://pubs.acs.org on August 17, 2017

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The Journal of Physical Chemistry

Correlating Free-Volume Hole Distribution to the Glass Transition Temperature of Epoxy Polymers Amin Aramoon,

†

Timothy D. Breitzman,

‡

Christopher Woodward,

El-Awady

†Department

‡

and Jaafar A.

∗, †

of Mechanical Engineering, Whiting School of Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA

‡Air

Force Research Laboratory, Wright-Patterson Air Force Base, OH 45433, USA

E-mail: [email protected] Phone: +1 410 516 6683. Fax: +1 410 516-7254

Abstract A new algorithm is developed to quantify the free-volume hole distribution and its evolution in coarse-grained molecular dynamics simulations of polymeric networks. This is achieved by analyzing the geometry of the network rather than a voxelized image of the structure to accurately and eciently nd and quantify free-volume hole distributions within large scale simulations of polymer networks. The free-volume holes are quantied by tting the largest ellipsoids and spheres in the free-volumes between polymer chains. The free-volume hole distributions calculated from this algorithm are shown to be in excellent agreement with those measured from Positron Annihilation Lifetime Spectroscopy (PALS) experiments at dierent temperature and pressures. Based on the results predicted using this algorithm, an evolution model is proposed for the thermal behavior of an individual free-volume hole. This model is calibrated such that the 1

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average radius of free-volumes holes mimics the one predicted from the simulations. The model is then employed to predict the glass-transition temperature of epoxy polymers with dierent degrees of cross-linking and lengths of prepolymers. Comparison between the predicted glass-transition temperatures and those measured from simulations or experiments implies that this model is capable of successfully predicting the glass-transition temperature of the material using only a PDF of the initial free-volume holes radii of each microstructure. This provides an eective approach for the optimized design of polymeric systems on the basis of the glass-transition temperature, degree of cross-linking, and average length of prepolymers.

Introduction The glass-transition temperature,

Tg , is the temperature at which a polymer transitions from

a glassy phase to a soft rubbery state, and it is one of the most important properties for any epoxy system. The fundamental mechanisms controlling this transition have been the subject of debate for many years.

Viscous ow inside a polymer network, as a thermally

activated process, has an important role in describing the underlying mechanism of this transition

13

. Given adequate time, viscous ow takes place by local rearrangements of single

chains over local potential barriers. This rearrangement, though not an irreversible process, severely retards the recoverable elastic deformation of the system

1,2

.

Thus, irreversibility

occurs when these local rearrangements take place in many regions in the polymer. As the temperature decreases the relaxing monomers form tightly packed groups in the amorphous states of the polymer. These packs will give rise to a distribution of local barrier heights, which require a temperature low enough to suppress motion over these barriers. The rise in the local barrier heights will hinder the viscous ow of the molecules in the system such that when the temperature approaches the glass-transition temperature the viscous ow can no longer occur at an observable rate.

This is where a glassy transition is observed.

However, the possibility of viscous rearrangements involving the sites with lower barriers

2


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still exists, which will give rise to a secondary relaxation if the temperature is held below the glass-transition temperature

1,2

.

The free-volume holes in a polymeric system, which are dened as the volumes between polymer chains within the microstructure, are typically characterized as spherical or ellipsoidal volumes between polymer chains in the network

4,5

. The size of these free-volume holes

can be directly correlated to the energy of their surrounding atoms. When these atoms are tightly packed, the energy elds in the core of these volumes are high. As a result, the energy required for the atoms to move into these volumes are high, and consequently, these atoms

6

are comparatively less mobile than those in the vicinity of larger free-volume holes .

In

addition, the structural features of the polymer network are important in understanding the mobility of the chains in the system. The polymer chains in a non-cross-linked system are more mobile than those in a highly cross-linked network. Therefore, the free-volume holes associated with cross-linked chains is smaller than those associated with non-cross-linked ones

58

.

The free-volume hole distribution in a polymer network can be measured experimentally using positron annihilation lifetime spectroscopy (PALS). The PALS experiments are based on the fact that annihilation of ortho-positronium (o-Ps) particles in low electron density regions of the structure, such as vacancies in the crystalline systems or free-volume holes in an amorphous one, is a very sensitive indicator of the size of these regions

5,7

. Since the

annihilation time of an o-Ps particle is a direct function of the size of the low density region that the particle ies through, the distribution of the size of these regions inside a system can be measured if the annihilation time of o-Ps particles created by the reaction of positrons with an electron cloud in its surrounding environment is formulated as a function of size of these regions

5,6,8

.

It has been shown that there is a direct correlation between the distribution of freevolume holes and the temperature/pressure response of polymers

5,8

. In addition, the glass-

transition temperature calculated from the average radius of free-volume holes was shown to

3



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be in good agreement with those measured from experiments such as dierential scanning calorimetry (DSC)

5,8,9

. Nevertheless, one of the short-comings of PALS experiments is that

they are unable to measure the location specic measurements of free-volume holes, which makes it impossible to correlate free-volume hole to local phenomena such as unrecoverable plastic deformation or crack nucleation. In addition, even with all experimental advances in quantifying free-volume holes, experimental investigations of the molecular mechanisms of polymeric materials still suer from complications and diculties associated with collecting conformation specic data

10

. Thus, this requires the utilization of physics-based numerical

methods that are able to address these limitations. Numerical methods such as molecular dynamics (MD) simulations have been extensively used to deduce the molecular details of plastic ow, glassy response, and the pressuretemperature dependence of polymeric systems

10,11

.

Many MD studies have also shown a

correlation between dierent polymer properties and the degree of cross-linking connectivity

12

, curing agent functionality

12

, and chain congurations

13

11

, network

. The rst attempt

in quantifying cavities and voids from MD simulation was by voxelizing the simulation cell, then tagging each voxel as a free or occupied cell to create a qualitative pattern of the cavity in the system

14

. In another study based on the work by Rigby et al., the cavities are

quantied by probing the volume with spheres having dierent radii then compared to those measured from PALS experiments

15

. This method was used in a series of studies to calculate

the free-volume size in atomic systems

1620

.

More recently, the method of voxelizing was

vastly optimized to calculate the free-volume in a microstructure at dierent temperatures for systems composed of thousands of polymer chains

21

. Nevertheless, the major drawbacks

of this method are the time and resources needed to analyze larger systems. For instance, voxelizing a simulation cell with an edge length of

20 nm

with a voxel size of

resources large enough to store information associated with

0.5 Å

requires

64,000,000 voxels, in addition to

the information associated with the location and size of the individual atoms in the system. Nevertheless, there is still a lack of understanding of the role of microstructure variations

4


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on the properties of polymers. Despite the success of MD simulations as a tool to illuminate the molecular origins of properties of materials, MD simulations suer from severe limitations in the accessible time and length scales. These limitations are more pronounced in simulations of polymeric systems where larger simulation cells are needed to model a statistically independent system in both spatial and temporal dimensions. One way to overcome these limitations is to reduce the number of degrees of freedom in the atomic structure by coarsening the atoms into a fewer number of coarser particles, called super-atoms, while properly averaging the high-frequency internal degrees of freedom of the molecule. This process lowers the number of degrees of freedom, while still capturing important physical properties of the material. Coarse-grained simulations also result in damping the high-frequency atomic vibrations, which dramatically shortens the eective time of simulations as compared to the identical fully atomic simulation. Furthermore, the problem of quantifying free-volume holes from coarse-grained simulations precludes the use of the voxelization method where the size of simulation cells are considerably larger. In this paper, we utilize large scale coarse-grained simulations to correlate the free-volume hole distribution to the glass transition temperature of epoxy polymers. In that attempt, a new method is developed to analyze the geometry of the polymer network rather than a voxelized image of the structure to accurately and eciently nd and quantify free-volume hole distributions within large scale simulations. The predicted free-volume hole distributions from the simulations are compared to those measured from PALS experiment to verify the model. Finally, a new thermal evolution model for free-volume holes are proposed to better predict the glass-transition temperature of the epoxy system for any given initial distributions of free-volume holes.

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Simulations Methodology The details of the coarse-graining method are only summarized here for completeness and are discussed in more details elsewhere

22

. The material chosen is an epoxy polymer composed

of Diglycidyl Ether of bisphenol A (DGEBA) monomers with 1,4 Diaminobutane (DAB) and N,N'dimethyl-1,6-,diaminohexane (DDH). The chemical structure of the DGEBA epoxy monomer and the DAB and DDH curing agent are shown in Figure 1. The atoms in the repetitive chemical units of the DGEBA monomer and DAB and DDH cross-linkers are mapped to coarser particles, called Super-Atoms, as shown in Figure 2. The interactions between these super-atoms are represented by spring-like bond, angular, and torsional potentials to account for all the conformationally relevant degrees of freedom of the system. These potentials are calculated by inverting the Boltzmann distribution functions of these coarsegrained coordinates

22

. Here, the distribution functions are calculated by sampling over the

free energy surface of these molecules using Quantum mechanics (QM) simulations

23

. Due

to the similarity in the atomic structures of DDH and DAB curing agents, the coarse-grained potentials calculated for DAB

22

are also used interchangeably for the DDH molecules. How-

ever, the DDH curing agents can only form two bonds to DGEBA monomers, while DAB curing agents bond with up to four DGEBA monomers. To model the interaction between non-bonded particles a Lennard-Jones (LJ) potential

1.519 kCal mol−1

tial well,

,

particles,

σ , is 5.732 Å.

is

of the system

22

24

is utilized. The depth of the poten-

and the nite equilibrium distance between two non-bonded

These values are the optimized values to correctly predict the density

.

In addition to dening the coarse-grained inter-particle potentials, it is also of great importance to create a realistic representative amorphous microstructure for accurate representative coarse-grained MD simulation. Here, we utilize a dynamic coarse-graining algorithm, also discussed in details in

22

, to create highly cross-linked epoxy microstructures. This al-

gorithm, unlike other curing algorithms in literature (e.g.

2527

), dynamically evaluates the

degree of cross-linking at dierent regions of the simulation cell during cross-linking, then

6


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Resin : (DGEBA)

Curing Agents: (DAB):

(DDH):

Figure 1: The chemical structure of the materials employed in the current simulations.

Figure 2: Mapping of: (a) DGEBA; (b) DAB; and (c) DDH molecule atoms into coarser super-atoms. The equivalent radius of each super-atom,

Rc

is also indicated.

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redistributes reactive prepolymers to regions with lower degrees of cross-linking.

In this

algorithm, the rate by which cross-linking is allowed between active pairs of super-atoms in dierent subregions is also controlled by comparing the degree of cross-linking in each subregion to the average degree of cross-linking in the entire simulation cell. If a subregion in the simulation cell has a high degree of cross-linking as compared to the average degree of cross-linking, then a slower rate of cross-linking will be enforced on this subregion as compared to subregions with lower degrees of cross-linking. These two modications allow the system to evenly redistribute cross-linkers to regions with higher chances of cross-linking. Therefore, a uniform degree of cross-linking is achieved throughout the entire simulation cell using this curing algorithm for any desired degree of cross-linking of the system. Furthermore, this algorithm allows for reaching higher degrees of cross-linking in larger simulation cells as compared to other curing algorithms

22

.

Numerical Quantication of the Free-Volume Hole Distribution In the following, a new methodology is developed to quantify the free-volume distribution and its evolution from atomistic simulations of polymeric networks. First, the total volume of a super-atom is calculated as the smallest sphere that encapsulates all individual atoms that construct the super-atom. The radius,

Rc ,

of the dierent super-atoms are shown in

Figure 2. Second, in the current methodology the quantication process of free-volume holes in the cross-linked regions of a polymeric network is dierent than that in a non-cross-linked regions of the system. This is explained in details in the following two subsection, and the methodology is available as an open-source software code on Github

8


28

.

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Free-Volume holes in cross-linked regions The free-volume holes within the cross-linked regions of the network are determined rst. This is achieved by using a geometry-aware graph algorithm that searches through the network to nd the volumes conned between the cross-linked chains. To nd each free-volume hole, rst, a triplet of consecutively bonded super-atoms is randomly chosen (c.f.

[ ABC

as

A

to

shown in Figure 3(a)). The shortest path of connected supper-atoms from super-atom super-atom

C

that does not pass through super-atom

B

is then identied and it along with

the randomly selected triplet are designated as a closed loop of connected super-atoms, as shown in Figure 3(b). This process is repeated for all available triplets of super-atoms until all the closed loops in the network have been identied. Each loop bounds a polygonal surface inside the network. These surfaces create envelopes that wrap around the free-volume holes inside the simulation cell.

In order to quantify these volumes, the largest ellipsoids

inside each wrap of polymer chains are tted, as shown in Figure 3(c).

(a)

(b)

(c)

Figure 3: Detecting free-volume holes in a cross-linked network: (a) a random triplet of connected super-atoms is chosen (e.g. ABC), (b) the shortest path from super-atom A to super-atom C that does not pass through super-atom B is determined to dene a closed loop polygonal plane; (c) the largest possible ellipsoid is then tted in the conned volumes between the connected polygonal planes.

Free-Volume holes in non-cross-linked regions One way to quantify the free-volume holes in a cluster of molecules is to create a Voronoi diagram of the system using the Delaunay tessellation algorithm

9


29

. Each Delaunay tetra-


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hedron is a tetrahedron that connects four atoms at its corners such that no other atom would be inside its circumsphere. Thus, each tetrahedron represents an isolated small freevolume hole.

The small isolated free-volume holes are then joined into larger free-volume

holes by merging neighboring tetrahedrons until the largest sphere that is centered at the center of geometry of the merged tetrahedrons is found. This is achieved by rst merging two neighboring tetrahedrons together, then calculating the minimum distance of the center of geometry of the merged tetrahedrons to the surface of each super-atom at the corners of these tetrahedrons.

If this distance is larger than the radius of the spheres tted in each

tetrahedron separately, then the two tetrahedrons are merged. Otherwise, two other neighboring tetrahedra will be tested. This process is repeated until all possible free-volume holes in the system are identied. To help speed up the analyses of the tetrahedron merging process, the following guidelines are enforced:

•

If the center of geometry of any tetrahedron is located inside one of the spheres dening the volume of a super-atom at one of the tetrahedron corners, then the tetrahedron is identied as being too small to contribute to the free-volume holes of the system and is ltered out of the calculations.

•

If the aspect ratio of the largest ellipsoid that can be accommodated inside the volume merged by two neighboring tetrahedrons is larger than a critical value (chosen here to be 15) then this merger is not allowed.

•

For any tetrahedron, the merging check is performed rst with the neighboring tetrahedron that shares its largest face. This is because the merging of a tetrahedron with its neighbor through its largest face will result in a volume that is larger than merging the same tetrahedron with one of its neighbors through a smaller face, as shown schematically in Figure 4(a).

•

If the angle formed between the lines passing through the center of geometries of two

10


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neighboring tetrahedrons to the center of their common face makes an acute angle (Figure 4(b)), the result of merging these two tetrahedra will be a distorted volume. Thus, this merging process will be skipped since one ellipsoid can not accurately represent the shape of this distorted volume.

It should be noted that an ellipsoid is the best representation for a highly irregular freevolume hole

21,30

. Thus, after all the volumes are created in the system, an ellipsoid is tted

inside each volume to provide a quantitative measure of that volume. These ellipsoids can be used to extract data such as the aspect ratio and the volume of the dierent free-volume holes. However, given that the quantum state of the positron is a sphere, the radii that are used for comparison with PALS experiments are the radii of the spheres t at the center of each ellipsoid.

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Figure 4: The process of analyzing the merger of a tetrahedron (green) with one of two neighbors: (a) The common face between the green and red tetrahedrons is larger than that between the green and black tetrahedrons. As such the ellipsoid that can be accommodated in the merged volume between the green and red tetrahedrons will be larger than that accommodated by the merger of the green and black tetrahedrons. (b) The angle formed between the lines passing through the center of geometries of the green and black tetrahedrons to the center of the merging face makes an acute angle, thus, the merging volume would be distorted. This merger is thus skipped. The angle between the green and red tetrahedrons is obtuse, thus, this merger is allowed.

Numerical Simulations All the numerical simulations performed here are conducted using the open-source MD code LAMMPS

31

.

The epoxy network is created by randomly populating a mixture of DAB,

DDH, and DGEBA super-atoms with a ratio of 2:3:5 in a cubical simulation cell having an edge length of

70

nm.

Periodic boundary conditions are employed along all directions

to mimic bulk properties of the epoxy network. these molecules at a density of

1.12 g·cm−3 .

The dynamic curing algorithm

to create highly cross-linked epoxy networks. linking, the microstructure is relaxed for

The initial system is a liquid mixture of

22

is employed

After achieving the desired degree of cross-

1,000,000 time-steps at the desired temperature with

12


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a time step of 5 fs. The simulations are performed using the NPT ensemble where the NoseHoover thermostat and barostat are used to control the temperature and pressure during the simulations. Temperature and pressure damping parameters are respectively.

200 and 3000 units of time,

The eect of temperature, pressure and microstructure features (e.g.

initial

length of prepolymers and degree of cross-linking) on the free-volume hole distribution are quantied by systematically varying one parameter and xing the others as detailed in later sections.

Results Temperature eects The eect of the temperature on the free-volume hole distribution is investigated for coarsegrained polymeric networks constructed from prepolymers having an initial length of 5 monomers and a degree of cross-linking of

85%.

The microstructure is rst relaxed at

and at a pressure of 0 Kbar. The temperature is then decreased to The distribution of free volume holes at at room temperature is measured density of The density,

ρ,

1.16 g·cm−3 ,

300 K

600 K

100 K in 4×106 time-steps.

is shown in Figure 5. The computed density

which is in good agreement with the experimentally

1.17 g·cm−3 7 . and the average radius of free-volume holes,

simulation at dierent temperatures,

T,

Γ,

computed from this

are shown in Figure 6. It is observed that there is a

strong correlation between the evolution of the average radius of free-volume holes and the density of the epoxy. As shown by the dashed lines in Figure 6, the average radius of freevolume holes in the range of temperatures simulated here can be divided into two dierent stages, which distinguishes the glassy state from the rubbery state of the epoxy. This is in agreement with experimental observations

32,33

.

Typically, the glass-transition temperature is calculated from the density or specic volume, versus temperature by tting two lines to the data in the glassy and rubbery stages.

13



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Page 14 of 30

Figure 5: The distribution of the free-volume holes at

300 K and 0 KBar for a coarse-grained

polymeric network constructed from prepolymers having an initial length of 5 monomers and a degree of cross-linking of

85%.

The colors are only to help with visualization.

The temperature at the intersection of the two lines is the glass-transition temperature. From the data in Figure 6(a), the predicted glass-transition temperature is

' 110◦C.

The

glass-trainsition temperature can also be measured from the relationship between average

7

radius of free-volume holes and temperature in a similar fashion . The glass-transition predicted from the evolution of free-volume holes as calculated from the current coarse-grained simulation is in good agreement with the one calculated from the density versus temperature curve, suggesting a strong correlation between the free-volume hole distribution and thermal evolution of the system

1618,20,34

.

14


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3.35 ρ Γ

1.2

3.3

1.16

3.25

1.12

3.2

Γ (Å)

ρ (g.cm−3 )

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3.15

1.08 200

300

400

500

600

Τ (K) Figure 6: The density and average radius of free-volume holes as a function of temperature from the coarse-grained simulation of the polymeric network at 0 KBar constructed from prepolymers having an initial length of 5 monomers and a degree of cross-linking of

85%.

Pressure eects The eect of the pressure on the free-volume holes is investigated for a coarse-grained polymeric network constructed from prepolymers having an initial length of 5 monomers and a degree of cross-linking of rates equal to

85%.

The cured system was cooled to

0.02 K ps−1 , 0.015 K ps−1 .

300 K at two dierent cooling

Then, the systems were relaxed at three dierent

pressures. The mean and the standard deviation,

µ,

of the free-volume holes radii at these

dierent pressures as predicted from the current simulations, and PALS experiments shown in Table 1.

34

are

A good qualitative agreement is observed between the simulation pre-

15



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dictions and PALS experiments.

Page 16 of 30

The standard distribution of free-volume holes becomes

tighter at lower pressures for both cooling rates, similar to the one measured from PALS experiment. While the average radius of the free-volume holes predicted from the current simulations are consistently larger than those measured from PALS experiments, it is clear that with decreasing cooling rate the predicted values from the simulations become closer to the experimental measurements. This can be attributed to the fact that the curing process and rate at which the cured epoxy is cooled in the simulation are drastically higher than those used for processing the epoxies used in the PALS experiments.

In addition, PALS

experiments sample the charge density and bonding distances realized in the polymer, while the current methodology of predicting the free-volume holes computes the largest spheres conned by the coarse-grained polymer chains without accounting for the Van der Waals forces between the dierent chains. This leads to predicting slightly larger free-volume holes sizes in the current simulations. Furthermore, the length of the prepolymers can also lead to variations between the predicted values from the current simulations versus experiments. Table 1: The mean and standard deviation of the free-volume holes at dierent pressures

kbar (Γ) µ (Γ)

p = 0 mean Experiments

34

kbar mean(Γ) µ (Γ) p = 1.8

kbar mean(Γ) µ (Γ) p = 4.5

2.4

0.31

1.95

0.28

1.5

0.15

Simulation at 0.02 K/ps

3.11

0.28

2.72

0.23

2.25

0.11

Simulation at 0.015 K/ps

2.95

0.29

2.63

0.25

2.24

0.13

Eects of prepolymer length and degree of cross-linking The eect of the length of the prepolymers on the predicted free-volume hole distribution is investigated for coarse-grained polymeric network constructed from prepolymers having initial lengths of 3, 4, and 6 monomers and a degree of cross-linking of

16


85%.

In addition,

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the eect of the degree of cross-linking on the predicted free-volume holes size distribution is investigated for coarse-grained polymeric network constructed from prepolymers having initial lengths of 5 monomers and a degree of cross-linking of both cases, the microstructure is rst relaxed at temperature is then decreased to

150 K

in

The free-volume holes average volume,

4 × 106

550 K

75%, 85%,

and

95%.

In

and at a pressure of 0 Kbar. The

time-steps.

χ = 43 πΓ3 ,

as a function of temperature for dif-

ferent degrees of cross-linking with an initial prepolymer length of 5 monomers are shown in Figure 7(a), while the relative change in the free-volume holes average volume with respect to that computed at 600K,

∆χ,

as a function of temperature for dierent lengths of

prepolymers and a degree of cross-linking of

85%

are shown in 7(b). Furthermore, the pre-

dicted glass-transition temperature from these simulations are summarized in Figure 8. It is clear that there is a strong correlation between the predicted glass-transition temperature and the average volume of free-volume holes with both the initial length of the prepolymers and the degree of cross-linking.

The average volume of free-volume holes decreases with

decreasing the initial length of the prepolymers and increasing the degree of cross-linking. Thus, the mobility of the polymer chains will also decrease, and the predicted glass-transition temperature will subsequently increase.

A Predictive Microstructural-Based Model of the GlassTransition Temperature As shown in the previous section, the free-volume hole size correlates strongly with the glasstransition temperature of the epoxy. As such, in this section the free-volume hole distribution is incorporated into a model to predict the glass-transition temperature of epoxy polymers. To that end, one can assume a polynomial function to represent the evolution of the size of a single free-volume hole as a function of temperature. This model can capture the evolution of free volume holes observed in simulations. The mathematical representation of this model

17



0 -5

Length of prepolymers: 3 Monomers 4 Monomers 5 Monomers

3

85

Degree of cross-linking: 35% 50% 75% 85% 95%

∆ χ (Å )

90

χ (Å3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 30

80 75

-10 -15

70

-20

65

-25 200

300

400

500

600

200

300

400

500

600

T (K)

T (K)

Figure 7: (a) The free-volume holes average volume as a function of temperature for dierent degrees of cross-linking with an initial prepolymer length of 5 monomers, and (b) the relative change in the free-volume holes average volume with respect to that computed at 600K as a function of temperature for dierent lengths of prepolymers and a degree of cross-linking of

85%. employed here, takes the form:

(1)

Γ = f (T ) = Pn (T )

where

Pn (T )

is an

nth

order polynomial function. Having an evolution of free-volume hole

radius as a function temperature, one can calculate the inverse function, such that for a given free-volume hole radius,

δΓ = δT where

αi

are the coecients

δg(Γ) δΓ

Γ,

g(Γ) = f (T )−1 ,

the following equation can be obtained:

−1 =

n−1 X

αi Γi

i=0

g(Γ).

18


(2)

Page 19 of 30

0.40 0.35 0.30 3

χ (nm )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.25 0.20 0.15 0.10 0.05 320

Different Lengths of Prepolymers Different Degrees of Cross-Linking

330

340

350

360

Τg (K) Figure 8: The free-volume hole average volume versus the predicted glass-transition temperature of the epoxy for dierent length of prepolymers and dierent degree of cross-linking. The simulations were conducted at a pressure of 0 Kbar.

19



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Page 20 of 30

Consequently, for any measured/predicted distribution of free-volume holes, the evolution of the free-volume holes average radius as a function of temperature can be computed by calculating the change in radius of each individual free-volume hole using equation (2). The values for the dierent

α's

should then be optimized such that the free-volume hole

average radius computed from averaging the evolution of the individual free-volume hole using Equation (2) would be equal to the real average radius of free-volume holes for the polymer. To calibrate the model, rst, the free-volume holes of a polymeric network constructed from prepolymers having an initial length of 5 monomers and a degree of cross-linking of are calculated at

300 K

35%

and 0 Kbar. The evolution of free-volume hole radii for each free-

volume hole is then is then separately calculated using Equation (2) by changing temperature from 150

K to 600 K with ∆T = 2.5 K.

The values of

αi

are optimized such that the average

radius of free-volume holes as computed from the evolution of each individual hole using equation (2) matches the one calculated from the simulation. The best t the simulated versus

T

χ

relationship are:

δΓ = δT ×

δg(Γ) δΓ

−1

= δT × 2.0489e−06 Γ2 − 3.71061e−4 Γ + 0.0037

(3)

The predicted free-volume holes average radius versus temperature from Equation (3) are also shown in Figure 9. It is clear that there is a good agreement between the model and simulation results. The thermal expansion coecients calculated from the model are also in good agreement with those calculated from the simulations. Given that the chemical potentials do not change with changing degree of cross-linking or lengths of prepolymers, Equation (2) with the optimized parameters in Equation (3) can now be used to predict the free-volume hole average volume versus temperature curves for the same epoxy with dierent degrees of cross-linking and lengths of prepolymers. This is shown

20


Page 21 of 30

Simulations Model 75

3

χ (Å )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


70

65

200

300

400

500

600

Τ (K) Figure 9: Comparison between the average volume of free-volume holes, from the model, and predicted from the simulations.

21


χ,

as calculated


in Figure 10 along with the results from the simulations (previously reported in Figure 7(a)) for comparison. It is clear that the model prediction of the glass-transition temperature of the polymer in each case matches well those predicted from the simulations for the dierent degrees of cross-linking and lengths of DGEBA prepolymers.

95 90

Degree of cross-linking: 35% 50% 75% 85% 95%

85 3

χ (Å )

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 30

80 75 70 65 60

200

300

400

500

600

Τ (K) Figure 10: Comparison between the average volume of the free-volume holes at dierent temperatures as calculated from equation (1), shown asby solids lines, and as predicted from the simulations for epoxies having distribution of free-volume holes, shown by symboles and dashed lines.

In another example to further assess the validity of Equation (1) in predicting the glasstransition temperature, the parameters of Equation (2) were also optimized according to results from a PALS experiment to see if it can reasonably predict the glass-transition temperature of the system given the distribution of free-volume holes. First, a large number of

22


Page 23 of 30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


free-volumes holes were sampled over the free-volume holes radii PDF at experiments

34

40 ◦C

from PALS

, as shown in Figure 11(a). These along with the average volume of holes as

a function of temperature (shown in Figure 11(c)) are used to optimize the parameters in equation (2), and the optimized values are:

δΓ = δT ×

δg(Γ) δΓ

−1

= δT × 1.4798e−6 Γ2 − 2.0095e−4 Γ + 0.0072

The PDF of the free-volume hole radii at

(4)

100 ◦C (Figure 11(b)) was then used to test the

model. The predicted curves of the average volume of free-volume holes versus temperature are plotted in Figure 11(c). The predicted curves show a good agreement with the experimentally measured curve, indicating that the model can successfully capture the evolution of free-volume holes as a function of temperature, and accurately predict the glass-transition temperature.

23



(b) 1.6

1.4

1.4

1.2

1.2

PDF (A-1)

(a)1.6

PDF (A-1)

1.0 0.8 0.6

1.0 0.8 0.6

0.4

0.4

0.2

0.2 0.0 2.4

0.0 1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

Radius (A)

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

4.2

Radius (A)

(c)

0.2 0.18

Experiment 40C Sample 100C Sample

0.16

(nm3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 30

0.14 0.12 0.1 0.08 0.06 0.04 -40

-20

0

20

40

60

80

100

120

140

T (C) ◦ (a) Probability density function of the free-volume holes radii at T = 40 C. ◦ (b) Probability density function of the free-volume holes radii at T = 100 C. (c) The ◦ experimentally measured curves, calibration curves at T = 40 C, and computed curves at ◦ T = 100 C of the average radius of free-volume holes versus temperature. Figure 11:

The above results suggest that equation (2) can be calibrated given an initial distribution of free-volume holes and the evolution of the average free-volume holes as a function of temperature for a given degree of cross-linking and length of prepolymers, then eectively used to predict accurately the glass-transition temperature for another polymeric network having the same chemical composition but dierent degrees of cross-linking or length of prepolymers. It should be noted that the optimized values are a function of the cooling rate. Therefore, it can only be utilized when the glass-transition temperature is required at the same cooling rate at which the model was calibrated to.

24


Page 25 of 30

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Conclusions A new algorithm was developed to quantify the evolution of free-volume hole distributions from the simulations of epoxy polymers. This method utilizes dierent algorithms based on whether the free-volume holes are conned in a cross-linked or an non-cross-linked region of the polymer. The characterization of the size and distribution of free-volume holes is then determined by tting the largest ellipsoids and spheres in each free-volume hole. To validate this algorithm, several benchmark simulations were conducted in which the free-volume hole radii distributions were compared to the ones measured in PALS experiments.

The

algorithm is shown to provide comparable results with experiments at dierent temperatures and pressures. The thermal behavior of an individual free-volume hole is then characterized using a model, which is calibrated to mimic the accurate average behavior of free-volume holes of a benchmark simulation/experiment. The calibrated model was employed to predict the glasstransition temperature of dierent microstructures by evolving every free-volume hole via the model. Comparison between the predicted glass-transition temperatures and the simulated glass-transition temperatures of dierent degrees of cross-linking and lengths of prepolymers implies that this model is capable of successfully predicting the glass-transition temperature of the material using only a single PDF of the free-volume holes radii of each microstructure. To further investigate this observation, the free-volume holes radii PDFs measured in PALS experiment for an epoxy polymer at two dierent temperatures were also used to sample two sets of free-volume holes. The free-volume holes sampled over the PDF at

40 ◦C

were

employed to calibrate the model for the material. The model was then tested using the freevolume holes sampled over the PDF at

100 ◦C

to predict the glass-transition temperature

of the material. The comparison between the predicted curve and calibrated curve shows a tremendous agreement, proving that the model is capable of predicting the thermal evolution of free-volume holes in an epoxy polymer. The proposed model can be utilized to eectively optimize the design of polymeric systems on the basis of the glass-transition temperature,

25



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Page 26 of 30

degree of cross-linking, and the average length of prepolymers.

Acknowledgement This work has been supported through a grant No.

FA9550-12-1-0445 to the Center of

Excellence on Integrated Materials Modeling (CEIMM) at Johns Hopkins University awarded by the Air Force Research Laboratory: the Air Force Oce of Scientic Research and the Materials and Manufacturing Directorate.

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Graphical TOC Entry

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 200

250

30


300

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Correlating Free-Volume Hole Distribution to the Glass Transition

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