Physically based modeling of shear modulus-temperature relationship for thermosets.
The material properties of polymers, such as the E-modulus, decrease with the temperature rise caused by the increasing mobility of polymer chains induced by the absorbed thermal energy. This decrease is more pronounced when the temperature reaches the glass transition temperature ([T.sub.g]). Several methods have been proposed for the estimation of the [T.sub.g] of thermoplastic polymers.
Dimarizo and Gibbs  and Gibbs and Dimarizo  introduced the thermodynamic method and assumed that the [T.sub.g] of polymers with simple molecular structures (dilute polymer solutions and thermoplastics) is the temperature at which the molecular entropy decreases to zero. The entropy is a derivation of the free energy of the system, and the latter is the product of the Boltzmann constant, the temperature and the configuration partition function. The configuration partition function is a function of the total number of configuration states of the molecular chains, which is counted by a permutation and combination method known as the Flory-Huggins counting process [1, 2]. A simplification of the Flory-Huggins counting process was proposed for counting the configuration of the polymer chains of dilute polymer solutions by Fowler and Rushbrooke  and Guggenhein , However, the divergence between the calculated results and the experimental data was observed to increase with the increase of the percentage of the cross-linking parts. The method in [1, 2] was extended for the estimation of the [T.sub.g] of lightly cross-linked rubbers by modifying the Flory-Huggins counting process and introducing several rubber-characterized equations , This method was not applied for the estimation of the [T.sub.g] of thermosets since the high cross-linking density and entanglements between different polymer chains in the structure of thermosets do not allow the derivation of the configuration partition function of such materials.
The free volume theory introduced by Fox and Flory  and Flory  assumes that the volume of a material can be divided into the molecular volume and the free volume, the volume around each molecule. The free volume theory was proposed for studying the liquid-glass transition of dilute polymer solutions and thermoplastics that are softened at temperatures around their glass transition temperature. The [T.sub.g] of such polymers could be estimated by the free volume method due to the significant increase of their free volume at temperatures around their [T.sub.g] [8, 9], The [T.sub.g] in this case can be effectively estimated by measuring the thermal expansion coefficient of the material [6, 9]. Few theoretical models based on the free volume theory have been proposed to estimate the [T.sub.g] of polymers. A semi-empirical equation was proposed by Rogers and Mandelkern  to estimate the [T.sub.g] of polymers with similar molecular structures. According to this equation, the difference between the [T.sub.g] of two polymers with the same molecular structure is proportional to the volume difference of these two materials at the liquid state. If the [T.sub.g] and the free volume in the liquid state of one reference polymer is known, the [T.sub.g] of all polymers with the same molecular structure could be determined. In addition, the classification of polymers according to their molecular structure is not self-evident, leading to errors as shown in . However, the application of the free volume theory in thermosets is still limited and it is suggested that there is a lack of physical reality of the free volume theory .
Another method, based on the relationship between the imaginary part of the G-modulus, G", and the temperature T, was proposed in [11-17], again for thermoplastics. According to this method, the [T.sub.g] of a thermoplastic polymer is defined as the temperature corresponding to the peak of G" under dynamic loading. A G"-T equation was suggested by Rouse  to estimate the shear modulus of dilute polymer solutions exhibiting few interactions between their polymer chains. The G" in this equation was a function of the viscosity that could be estimated by the entropy of the material, and the mobility of the molecular chains. It was pointed out by Rouse  that the disagreement between the G" ([G.sub.2] in the paper) deduced from this equation and the experimental value was due to the length of the molecular chains of the actual polymers and the inter- and intramolecular interface. The expression of a critical parameter in the deduction of the G"-T equation (entropy) was extended to lightly cross-linked thermosets (rubber) by Mooney  under the assumption that the strands of rubbers (molecules between the cross-linking points) are equivalent to the single molecule of thermoplastics. A G"-T equation was for the simulation of the behavior of polydimethylsiloxane (PDMS) at a specific physical state during the curing process, the gel point, was derived by Winter and Chambon , This relationship was not appropriate for thermosets since PDMS at gel point was still far from being considered as a fully cross-linked material. A G"-T equation for reversible cross-linked polymers with temperature was proposed by Tanaka and Edwards  emphasized to the simulation of the decreasing rate of G" with the breaking of the primary bonds. An equation concerning the effect of entanglement on the shear modulus of natural rubber, was proposed by Everaers et al , however, only being applicable for the description of the mechanical response of the non-cross-linked region of the examined material. The accuracy of the above mentioned G"-T equations for thermosets are not satisfactory due to the ignorance of the effect of the cross-linking and the entanglements on the expression of the equation parameters.
A modified G"-T equation based on the equations in Ref. 11 is proposed in this work for the modeling of the loss modulus of thermosets during temperature elevation. The method introduces two new formulations regarding the calculation of the configuration probability and the velocity matrix. Appropriate formulations for the investigation of the influence of these three parameters on the loss modulus are developed, taking into account the cross-linking effects. The experimentally derived steady-flow viscosity vs. temperature relationship and an assumed sub-molecular mean square separation vs. temperature equation are necessary for the model calibration. The derived new G"-T equation is validated by modeling the behavior of a commercial structural epoxy adhesive (Sikadur-330, Sika AG).
G"-T MODEL FOR THERMOSETS
The development of the formulations describing the influence of the configuration probability and the velocity matrix is presented in this section.
Probability of Configuration
A schematic representation of the molecular structure of thermosets is shown in Fig. 1. Each sub-molecule in a molecular structure is linked with two others in a certain geometrical configuration. The probability ([P.sub.i]) that the end-to-end distances of the three sub-molecules (i- 1 ~i, i~/+1 and i~N/2+i) located around the representative point, i, on the molecular chain are [d.sub.i-I,i], [d.sub.i-i,i] and [d.sub.i,N/2+ i] respectively, see Fig. 2, is given by:
[P.sub.i] = [P.sub.i-I,i] x [P.sub.i-i,i] x [P.sub.i,N/2+i] (1)
as a product of the individual probabilities [P.sub.i-1,i], [P.sub.i,i+i] and [P.sub.i,N/2+i] of the end-to-end distance of each sub-molecule having the value [di-I,i], [d.sub.i, i+1] and di, N/2+i respectively.
It is assumed that the sub-molecules (i- 1~i, i~+1, i~N/2+i) are Gaussian chains . In addition, for simplification, all the angles between the end-to-end directions of adjunct polymer sub-molecules are equal to 0, see Fig. 3, the probability P that the end-to-end distance is d is :
P = [(b/[pi]).sup.3/2] exp(-[bd.sup.2]) (2)
where b is equal to 3/(2[[sigma].sup.2]) and [[sigma].sup.2] is the mean square distance (separation) of the ends of sub-molecules ,
After the movement of a representative point, from (0, 0, d) to (x, y, z), see Fig. 3, the end-to-end distance of each submolecule is given by:
[d.sup.2.sub.i-1,1] = [x.sup.2] + [y.sup.2] + [z.sup.2] [d.sup.2.sub.i,i+1] = [x.sup.2] + [(y - d sin[theta]).sup.2] + [(z - d - d cos [theta]).sup.2] [d.sup.2.sub.i,N/2+i]] = [x.sup.2] + [(y - d sin[theta]).sup.2] + [(z - d - d cos [theta]).sup.2] (3)
The [P.sub.i] of thermosets is obtained after discarding the first order expression of x, y and z,
[P.sub.i] = [P.sub.i -1,i] x [P.sub.i,i+1] x [P.sub.i,N/2+i] = [(b/[pi]).sup.3/2x3] exp (-b[d.sup.2.sub.i-1,i]]) exp (-b[d.sup.2.sub.i+1]]) exp (-b[d.sup.2.sub.i,N/2+i]) = [(b/[pi]).sup.9/2 exp(-b(3[x.sup.2] + 3[y.sup.2] + 3[z.sup.2])) (4)
And therefore, the probability of configuration of a molecular structure, like the one shown in Fig. 1, can be calculated by:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
The velocity vector of thermosets is also different from that of thermoplastics. The velocity of the representative point, i, is affected by the cross-linking and can be calculated if the velocity of the adjacent points is also considered.
The velocity of the representative point on the molecular chain of thermosets under dynamic loading is proportional to the applied driving force that can be estimated from the gradient of the free energy (the free energy is the product of the temperature (7) and the system entropy (5), -TS) along the corresponding coordinate direction . The relative velocity of the representative point i with respect to the adjacent points is given by:
[([partial derivative][x.sub.i]/[partial derivative]t).sub.i-1] = BT ([partial derivative]S/[partial derivative][x.sub.i] - [partial derivative]S/[partial derivative][x.sub.i-1]) (6)
([partial derivative][x.sub.i]/[partial derivative]t).sub.i+1] = BT ([partial derivative]S/[partial derivative][x.sub.i] - [partial derivative]S/[partial derivative][x.sub.i+1]) (7)
[([partial derivative][x.sub.i]/[partial derivative]t).sub.i+N/2] = BT ([partial derivative]S/[partial derivative][x.sub.i] - [partial derivative]S/[partial derivative][x.sub.i+N/2) (8)
where [x.sub.i] is the displacement of the ith representative point, t is the time, and B is a parameter representing the mobility of a sub-molecule.
The total velocity of the representative point is given as the sum of the relative velocities calculated above:
([partial derivative][x.sub.i]/[partial derivative]t) = BT ([partial derivative]S/[partial derivative][x.sub.i-1] + 3 [partial derivative]S/[partial derivative][x.sub.i]) - [partial derivative]S/[partial derivative][x.sub.i+1] - [partial derivative]S/[partial derivative][x.sub.i+N/2]) (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
The velocity matrix (VO, showing the velocity of all representative points on the molecular chain, can then be estimated by Eq. 10 ,
(where [A.sub.0] is the coefficient matrix for thermosets, introduced in this work,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
that is different from the coefficient matrix of thermoplastics, see , The eigenvalues of matrix [A.sub.0] are:
[[lambda].sub.1p] = 4[sin.sup.2] p[pi]/(2(N/2+1)) p = 1,2....N/2 (12)
[[lambda].sub.2p] = 2 + 4[sin.sup.2] (p - N/2)[pi]/(2(N/2+1)) p = N/2 + 1, N/2 + 2.... N (13)
Modified G"-T Equation
Based on Eq. 5, and Eqs. 12, 13 and the continuity equation, , the imaginary part of the G-modulus of thermosets is,
G" = [([square root of 3]2[[sigma].sup.2][pi]).sup.3N] nKT/3 [N.summation over (p=1)] [omega][[tau].sub.p]/1 + [[omega].sup.2] [[tau].sup.2.sub.p] (14)
with the relaxation time of the molecule [[tau].sub.p] given by:
[[tau].sub.p] = 72[[eta].sub.T]/nKT[[lambda].sub.1p ]N/2 (N/2 + 3) [(2[[sigma].sup.2][pi]/[square root of 3]).sup.3N], p = 1, 2.... N/2 (2) (15)
[[tau].sub.p] = 72[[eta].sub.T]/nKT[[lambda].sub.2p] N/2 (N/2 + 3) [(2[[sigma].sup.2][pi]/[square root of 3]).sup.3N], p = N/2 + 1, N/2 + 2.... N (16)
n = [[rho].sub.0]/M [N.sub.A] (17)
where [omega] is the loading frequency, k is the Boltzmann constant, n is the number of molecules per unit volume, [[eta].sub.T] is the steady-flow viscosity at temperature T, [[rho].sub.0] is the density of the material, and M is the molecular weight that can be estimated by: M = (N - 2) x [M.sub.resin] + (N/2) x [M.sub.hardener], and [N.sub.A] is the Avogadro constant.
The parameters in Eq. 14, except [[eta].sub.T] and [[sigma].sup.2], can be estimated according to the cross-linked molecular structure of thermosets consisting of molecules of the resin and the hardener. The chemical formulations of the molecules of these ingredients and the corresponding reactions can be found in technical data documents. The steady-flow viscosity, [[eta].sub.T], should be obtained by low-frequency dynamic mechanical analysis. The parameter [[sigma].sup.2], the mean square separation of the ends of sub-molecules, varies with temperature due to the different end-to-end distances of the sub-molecules that change because the molecular mobility increases with temperature elevation. In previous works, e.g. , [sigma] was assumed to be proportional to [N.sup.0.5.sub.0], where [N.sub.0] is the number of half of the sub-molecules along the length direction of the molecular chain of thermosets. An arctangent function was introduced here to simulate the effect of temperature on the variation of [sigma], which was assumed to be significant during the glass transition and the rubbery state and negligible in the glassy state of the thermoset, see Eq. 18 and Fig. 4:
[sigma](T) = q[N.sub.0.sub.0.5 + arctan] (- (T - [T.sub.0]/[T.sub.a]) (18)
where q is a fitting parameter, [T.sub.0] is the reference temperature at the rubbery state, and N0 is equal to N/2-1 according to the molecular structure shown in Fig. 1.[T.sub.a] = 1[degrees]C.
MODEL PARAMETER ESTIMATION AND VALIDATION
Estimation of Viscosity and q Parameter
Sikadur-330 is a thixotropic bi-component commercial structural adhesive produced by Sika Schweiz AG. The base resin is a bisphenol-A-based epoxy and the main hardener is trimethylhexane- 1,6-diamine, as shown in Table 1. The mixing ratio is 4:1 resin to hardener . Specimens with dimensions 53 mm x 10 mm x 3 mm were fabricated and used for the experimental investigation according to ASTM E1640-09 . In order to exclude the effect of post-curing during the experiment, the epoxy specimens were cured at ambient temperature for 48 h and then post-cured at 63[degrees]C for 144 h to approach a curing degree of 100% (according to ). The [T.sub.g] of fully cured Sikadur-330 was measured by DMA experiments at 1Hz and 1 [degrees]c/min. The value corresponding to the loss modulus curve is equal to 64[degrees]C.
The low-frequency dynamic experiment was performed using a Q800 DMA machine and a three-point bending fixture to estimate the viscosity [[eta].sub.T].
The low-frequency dynamic experiment was performed using the multi-frequency model of the Q800 at a loading frequency of 0.02Hz. A reversed cyclic displacement with amplitude of 15 pm was applied and the corresponding load was measured to calculate the loss modulus of the material. The specimen was heated up to 90[degrees]C at 0.2[degrees]C/min to minimize the thermal lag between the specimen and the surrounding environment inside the Q800 chamber.
The viscosity ([[eta].sub.T]) deduced from the low-frequency dynamic experiment can be estimated by the experimentally obtained loss modulus (E") of the specimen during DMA via the following equation:
[[eta].sub.T] = E"/(1+v)[omega]
where v denotes the Poisson's ratio of the examined epoxy resin, and in here has been assumed to be equal to 0.44, the default value designated by TA instruments . The viscosity derived from the low-frequency dynamic analysis as a function of temperature is shown by a solid line in Fig. 5. The viscosity initially increases with temperature, reaches a peak near the glass transition temperature and then decreases. The maximum estimated value of viscosity was 1250 MPa x s.
The molecular weight of the resin and the hardener can be calculated based on the chemical names of their components and the corresponding chemical structure. The concentration and the density were obtained from  and their values are shown in Table 1. An iterative trial and error process was used to estimate the value of the q parameter based on the abovementioned determined parameters. According to this process, the value of q was increased with an interval of 0.1% from an initial given value and a corresponding [sigma] value was calculated by Eq. 18. The appropriate q value (q = 13.3%, see Fig. 6) is the one that results in a [sigma] value which, when substituted in Eq. 14, provides a G"-T curve with a maximum at a similar temperature to that of the curve experimentally derived by a DMA experiment at 1Hz and a rate of l[degrees]C/min, see the dashed line in Fig. 6.
G"-T Model Validation
The comparison between the G"-T curve obtained by Eq. 14 and the experimentally derived one is shown in Fig. 7. The two curves coincide at the range of temperatures above [T.sub.g] , while the modeled G" is higher than the experimental value for temperatures below [T.sub.g] , reaching a maximum difference of ca. 70% at 35[degrees]C in the glassy state of the examined thermoset. The cross-linked molecular chains of thermosets are further linked to each other by entanglements and strong secondary bonds especially at temperatures below [T.sub.g] . The proposed molecular structure, shown in Fig. 1, and the assumed [sigma]-T equation were incapable of accurately simulating the behavior of the material at this stage, leading to the observed diversion between the two curves. During temperature elevation, the secondary bonds break and the entanglements between polymer chains loosen, so that the cross-linking molecular structure can be well described by the G"-T model provided by Eq. 14.
In this work, the G"-T relationship for thermoplastics has been modified to provide a physically based G"-T model for thermosets. All the necessary parameters for the model derivation, except one fitting parameter for the description of the mean square separation of the ends of the sub-molecules with temperature, were estimated based on the molecular structure of thermosets and low-frequency dynamic experiments. The fitting parameter was calibrated based on the experimentally derived G"-T curve of the examined thermoset.
New physically based formulations for the calculation of the configuration probability and the derivation of the velocity matrix of thermosets have been introduced in this work. The formulations were derived based on the molecular structure of thermoset resins taking into account the cross-linking and entanglements that such materials present.
An arctangent function between [sigma] and T was found to be appropriate to consider the different effects of temperature on the end-to-end distances of the polymer chains at the glassy state, the glass transition and the rubbery state of the thermosets.
The G"-T curve obtained from the introduced model corresponded well with the G"-T curve experimentally derived by a dynamic mechanical analysis, especially for temperatures above the glass transition. The divergence between the G"-T curve derived from the model and the experimentally derived one below [T.sub.g] is attributed to the strong link between the polymer chains at the glassy state that cannot be precisely described by the model.
A more complex cross-linking model is necessary in order to accurately simulate the G"-T curve at the glassy state comprising additional parameters for the description of the entanglement variation and the effect of the secondary bonds on the mobility of the polymer chains for temperatures below the glass transition temperature. By introducing such parameters, Eq. 14 can be appropriately modified to accurately predict the G"-T curves of thermosets.
NOMENCLATURE PDMS polydimethylsiloxane [A.sub.0] coefficient matrix of V of the modified G"-T method B mobility of the end of a sub-molecule b molecular structure constant d end-to-end distance of a molecule E" loss modulus G" imaginary parts of the complex shear modulus i ith representative point k Boltzmann constant (1.38 X [10.sup.-23] [m.sup.2] x kg x [s.sup.-2] x [K.sup.-1]) M molecular weight of the molecule of the polymer [M.sub.hardener] molecular weight of the hardener [M.sub.resin] molecular weight of the resin [N.sub.0] number of half of the sub-molecules along the length direction of the molecular chain of thermosets N number of representative points of each molecule [N.sub.A] Avogadro constant (6.02 X [10.sup.23] [mol.sup.-1]) n number of molecules per volume of thermosets P pth eigenvalue P probability that the end-to-end distance of a molecule on the polymer chain is d [P.sub.i] probability that the representative point (referring to the second end of the molecule) of the ith sub-molecule lies within the volume element [dx.sub.i][dy.sub.i][dz.sub.i] at the location [x.sub.i], [y.sub.i]. [z.sub.i] relative to the first end [P.sub.N] probability of the configuration where the representative points of the N sub-molecules lie within the volume element [dx.sub.1][dy.sub.1][dz.sub.1] ... [dx.sub.N][dy.sub.N][dz.sub.N] at the relative location ([x.sub.i] [y.sub.i], [z.sub.i])... ([x.sub.v], [y.sub.v], [z.sub.N]) q Model parameter S entropy of the molecule T temperature [T.sub.0] reference temperature at the rubbery state of the thermoset [T.sub.g] glass transition temperature t time V vector velocity of the representative points of polymer molecules [x.sub.i] displacement of the ith representative point[eta] [[eta].sub.T] steady-flow viscosity of thermosets [theta] angle between the end-to-end direction of adjunct polymer sub-molecules [[lambda].sub.1p], eigenvalues of the coefficient matrix of the [[lambda].sub.1p] velocity of the modified G"-T method v Poisson's ratio [[rho].sub.0] density of the material root of the mean [sigma] square separation of the ends of sub- molecules [[tau].sub.p] relaxation time of the molecule [omega] load frequency
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Wei Sun, Anastasios P. Vassilopoulos, Thomas Keller
Composite Construction Laboratory (CCLab), Ecole Polytechnique Federale De Lausanne (EPFL), Station 16, Batiment BP, Lausanne CH-1015, Switzerland
Correspondence to: A.P. Vassilopoulos; e-mail: anastasios.vasilopoulos@ epfl.ch
Contract grant sponsor: Swiss National Science Foundation; contract grant number: 200021_129613; contract grant sponsor: SIKA AG, Zurich.
Published online in Wiley Online Library (wileyonlinelibrary.com).
TABLE 1. Material parameters of Sikadur-330. Ingredients (a) Part A (Resin) Bisphenol A-(epichlorhydrin) epoxy resin 1,4-bis (2,3 epoxypropoxy) butane Other solvent Part B (Hardener) Trimethylhexane-1,6-diamine Other solvent Concentration weight (a) (%) Molecular (g/mol) (b) Part A (Resin) 70 356 289.6 ([M.sub.resin]) 20 202 10 Part B (Hardener) 75 158 1 18.5 25 ([M.sub.hanlener]) Density [[rho].sub.0] (g/[cm.sup. 3]) (a) N (b) Part A (Resin) 1.3 10 Part B (Hardener) (a) Data from Ref. 18. (b) Calculated data.
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|Author:||Sun, Wei; Vassilopoulos, Anastasios P.; Keller, Thomas|
|Publication:||Polymer Engineering and Science|
|Date:||Dec 1, 2015|
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