Simple quantification of elastomers’ deformation-induced entropy changes through maxwell relation implementation

An upper division experimental method to determine the configurational entropy change using an approximated Maxwell relation for short fiber reinforced polymers is presented. This approach mainly integrates two regression models in data analysis, respectively being the phenomenological Mooney-Rivlin model and the linear tension-temperature relation; the former simulates the tensile performance under various isothermal conditions, while the latter is used to measure the partial differential value ∂S∂lT in the Maxwell relation. To further correct tensile data to accurately measure the minute configurational entropy change in loading, a linear-form segmented compensation function was used to correct the dynamic fatigue effect in repetitive tensile testing. With corrected Mooney-Rivlin fits under various temperature conditions graphed, one may focus on a fixed strain and investigate its corresponding linear tension-temperature relation to establish the numerical value of ∂S∂lT. In this way, the remaining unknown variable in the Maxwell relation is entropy, thereby enabling its mathematical determination. Such a simple experimental design serves as a convenient method to measure configurational entropy change with strain for basic research.


Introduction 1.Motivation
To study how the formation of a network chain determines the tensile and thermodynamic properties, a number of theoretical analyses [1][2][3][4][5][6] and numerical simulations [7][8][9][10][11][12][13] have investigated the length or molecular weight of the monomer chains and the topology of networks.The two main theoretical models used to describe elastomer thermodynamics are affine [12] and phantom network models [13].In the affine model, the conjunction between successive monomers is assumed to be fixed in space.In contrast, the phantom network model assumes that the degree of vibration is independent of macroscopic properties; the corresponding expression of Helmholtz free energy derived from the Boltzmann entropy of these two network models thus differ by a factor of 1/2 and serves as an upper and lower boundary of entropy change during stretch or contraction.Furthermore, factors such as network defects (dangling chains, loops, trapped entanglements, etc) and, non-zero energy contribution to the elastic retractive forces render simulations with these two theories yield inaccurate values for free energy and configurational entropy change.Therefore, measurement-based approaches should be the primary approach for entropic analysis.
Previously, the measurement of f f s [14], which resembles the ratio of the entropy component of tension to equilibrium tension, took into account of the changing internal energy with elongation and thus improves the accuracy of network models; complicated procedures to determine the f f s for a wide range of strains and temperatures were implemented.Specifically, the experimental procedure to determine f f s requires sophisticated controls to measure minute tension changes over varying temperature, and further processing of key variables such as C 1 in the Mooney model, a less developed form of the Mooney-Rivlin model [15], to satisfy the molecular models for the measurement of entropy change.Therefore, a method using the Maxwell relation [16] which describes the partial differential of equilibrium tension f to the absolute temperature T with that of entropy S to, the length of elastomer l is applied as the fundamental to analyze the tensile tests performed at constant T and hence derive the entropic expression of elastomer.Rather than the Mooney model, the advanced Mooney-Rivlin model, is used to fit the tensile data.
Specifically, the use of the Maxwell relation with the aid of the Mooney-Rivlin model to determine configurational entropy change is direct and simple in several ways.
(1) While the equilibrium tension simulated with the phenomenological Mooney-Rivlin hyperelastic model allows for direct correction upon fitting [16], features such as random tensile behavior at low strain [17,18] and strain softening [19] at high strain do not require further calibration.
(2) The Mooney-Rivlin form provides a molecular basis grounded in statistical mechanics for entropy measurement [20], rendering further processing to fit the tensile data with the chain models dispensable.
(3) Using the Maxwell relation, the tension to temperature data may be approximated by a linear regression.

S l T
may be effectively obtained by relatively few tensile data fits, resulting in lower experimental complexity.

Methodology 1.2.1. Calculation of configurational entropy change using the approximated maxwell relation
The Maxwell relation in the form of formula (1), is derived from the thermodynamic identity in the Helmholtz free energy form: SdT 2 which describes the relationship between the Helmholtz free energy F and the variables in the Maxwell relation.
In the case of Helmholtz free energy, its second mixed partial derivative may be assumed to be continuous, so we have: Similar derivation was confirmed by Allen Wasserman for the Paradigms in Physics program at Oregon State University [21].Further details of the Paradigms in Physics curriculum are provided on the project's webpage [22].Formula (3) implies that the entropy changes during elongation at a constant temperature could be obtained by measuring the tension-temperature relation at a fixed length.The configurational entropy change can then be integrated as follows: In the elucidation of f f s method, the equilibrium tension consists of entropy and energy components.In formula (5), two partial derivatives representing the two components could be seen as constants, respectively being γ and β.Therefore, the tension in Maxwell relation can be represented as linearly related with temperature, with the form given by formula (6).
By connecting formula (4) and (6), we obtain formula (7), which suggests that the values of ( ) a best fit l up to elastomer length l may be numerically integrated to yield DS from relaxed state to l, with small step size κ.
To obtain the values of simulated ( ) a best fit l from the tension versus temperature data, the tensile tests are performed at multiple temperatures and the data is fitted with a hyperelastic model.

Mooney-rivlin constitutive model
Since a best fit from Formula (5) to (7) is obtained under controlled strain, the only way to apply Formula (7) is by numerical integration.During such process, the fitting step of ( ) a best fit l must be controlled to be small for accuracy.The most convenient way to enable the small fitting steps in measurement is to fit data from a model which contains temperature, strain, and tension as variables.By fitting the tensile data under various temperatures, the tension under fixed strain and temperature may be determined for the entire range of strains stretched.Under strictly controlled conditions (see 2.1, 2.3.2,2.4), a rearrangement of the variables of interest (from tension-strain to tension-temperature) would allow for the determination of the values of a best fit under a range of strains.
Moreover, the state of elastomers is highly unstable at low strain [17,18], and the effect of strain-softening tends to develop in high strain regimes [19,23].Thus, applying a hyperelastic material model aids to remove noise coverage and experimental uncertainty in measurement, so the values of tension obtained through the calculation with models tend to be more accurate than the direct measurements at extreme strain regimes.
While the tensile properties of elastomers are often described by Hooke's Law and Young's Modulus, Young's modulus is not applicable to large-scale elongation (0% ∼200% strain in experiments) and does not include the variable of temperature.To describe the tension-elongation relationship elastomer more accurately, continuum mechanics treatment should be implemented to account for the nonlinearity pattern within the elastomer mechanism.Moreover, to satisfy the determination of configurational entropy change via Maxwell relation, the hyperelastic model must also include temperatures as a variable.
As Natural Rubber fibers satisfy the properties of the Mooney-Rivlin solid, such as incompressibility, isotropy, and hyperelastic [23][24][25][26], its tensile behavior may be fitted with the Mooney-Rivlin model, a phenomenological theory of elastomer elasticity assuming a mathematical form for the constitutive equations of elastomer as a function of the three invariants of deformation, unified in formula (8) and (9).Since the simulated Mooney-Rivlin model in this study doesn't contain a turning point, it can be expressed as the first power function of I 1 and I 2 .

( )
Formula (10) is the fundamental form of the Mooney-Rivlin model, which describes the strain energy density W with the stretch ratio in three orthogonal principal directions l l l , , .
V 0 , l 0 , and A 0 in the derivation are respectively the volume, initial length and the initial cross-sectional area to stretch axis of the elastomer of studying.Furthermore, compared with the form of the model listed by Flory and Erman [27] which has the same prediction trend and performance as the Mooney-Rivlin model, we may assume that the coefficients in the Mooney-Rivlin model should be functions of temperature.Similar conclusion could be arrived at by comparing experimental data with the result of the Gaussian model.Therefore, the Mooney-Rivlin model could be perceived as a function of temperature, strain, and tension.
respectively represents the free energy density, the polymer chain contribution energy density.ξ, k, N, Bi, Di are all constants for fixed elastomer, and T represents absolute temperature.Therefore, W in the phenomenological simulation of Mooney-Rivlin model may be interpreted as a function of absolute temperature.
Given the three inherent variables in Mooney-Rivlin model (tension, temperature, and strain), tensiontemperature relationship may be studied for controlled strain after several Mooney-Rivlin fits at various temperatures are obtained.After linear tension-temperature relationship is assumed, the value of ( ) a best fit l could be obtained by measuring the gradients of the fitting functions for tension and temperature.

linear-form segmentation compensation function
Despite the accuracy of Mooney-Rivlin fits and carefully controlled experimental conditions, the fatigue effect is an unneglectable phenomenon for materials under repeated periodic loads.Although the static fatigue effect could be reduced to inconsequential by avoiding very high tensions and maintaining a relaxed state between tension measurements, the dynamic fatigue effect cannot be avoided in repetitive stretching [28], which gives rise to systematic error.Therefore, a linear-form segmented compensation function is deduced to correct the data of tension.
The form of compensation depends on the individual's materialistic properties.For high-elastic materials such as rubber, Bartenev and Ogly [28] suggests a linear relation between tension and time on a log-log diagram within the maximum 2500 loading cycles; as the stretching frequency in their study is held constant, it generalizes to a linear relation between tension and loading cycles.
Further consideration of the entropic basis in molecular chain interactions leads to the inclusion of a dynamic fatigue parameter D, evolving from 0 to 1, in tension form Mooney Rivlin expression.Treloar [29] plotted different loading cycles for his experiment on slightly vulcanized rubber, which shows that up to a stretch of about λ = 4.8, the material behavior is nearly ideally elastic.For strains below 100%, the plastic deformation would be inconsequential, and thus the incompressibility and sheer effect of dynamic fatigue effect may be still assumed, rendering the application of Formula (13) suitable.
Formula (13) can be further simplified by assuming a linearly degrading pattern of D, which is applicable for mimicking the stiffness loss due to the progressive progress of bond pole movement or microscopic damage (which doesn't contribute to plastic deformation) in short fiber reinforced polymers.Further assumption that temperature bears insignificance upon the dynamic fatigue effect [30] provides ground for application of a single compensation function amongst all temperature conditions, described in two forms: where ε is the strain, ∆f comp is a constant for each rubber, and N is the cycle number.The latter form is based on the assumption that the dynamic fatigue parameter is a function of strain within certain strain intervals, allowing for the fitting of Mooney-Rivlin at low strain, owing to the passing through origin, and the variation in strain softening behavior between two tensile datasets varying over a large interval of the loading cycle.Formula ( 14) is used to compensate for the tensile data points by adding attenuate forces to the measured value based on the compensation function in that range and the current cycle number.Subsequently, the corrected Mooney-Rivlin fit are simulated and used to determine the values of ( ) a .The schematic of the experimental setup is shown in figure 1.During stretching, the rubber bands are immersed in water in an acrylic tube, which was held with a lifting support on an electronic scale.The following is an introduction of the experimental apparatus in the given sequence in figure 1.
(1) Electronic scale: A high-precision electronic scale is utilized to enable both continuous and precise measurement of small tensions.
(2) Lifting support: A lifting support is tied to the upper end of rubber band and placed on the electronic scale to keep the acrylic tube in stable horizontal position and the rubber band on the uniaxial vertical route to reduce uncertainty due to left-right movement during stretch.It also aids control for stretching pace and hence the mechanical equilibrium to render tension equal to scale recording.
(3) The acrylic tube and wooden stopper with implanted hook: During stretch, the rubber band is hooked to stay in place inside the acrylic tube, which functions as a heat bath to control the temperature before and after loading.Based on elastomer's isotropic nature, strain is obtained by measuring elongation of the length below the dotted point on rubber and then dividing the effective length (length above the hook).
(4) Rubber: The rubber bands applied in this study are the Natural Rubber fibers, which fall within the class of short fiber reinforced polymers and are similar to those used by Ogly [28].Thus, they are assumed to largely satisfy the linearity in compensation function.

Data processing software
Rather than measuring the strain manually using a ruler external to the tube, a simultaneous video analysis approach was employed to record at horizontal view (adjusted using a spirit level).
As shown in figure 2, using a scale bar and the video image magnification on Tracker, the strain and corresponding tension of the rubber band were conveniently measured and used to produce Mooney-Rivlin fits.

Experimental objective
In this study, the configurational entropy change of Natural Rubber fibers in the stretching process is studied.

Experimental design 2.3.1. Preliminary experiment
In the preliminary experiment, the tensile tests for rubber of all widths were set at 330K and a strain amplitude of 200% to compare the dynamic fatigue effect.The latter requirement of a large strain amplitude not only improves the accuracy of the approximated linear fit, but also reduces the possible noise coverage in the larger  picture of overall tension-strain dataset, which, for rubbers with stronger tensile behavior, is eventually capable of overlapping dynamic fatigue in a low-strain regime.
To obtain the ∆f comp in the compensation functions, data between the 1st and the 10th, 20th, 30th, and 40th experiments were processed, and the compensation function's coefficients of latter four were averaged to obtain general compensation function.This is because, even after several improvements in the experimental design, the dynamic fatigue accumulated between short intervals of loading is too subtle to be recorded precisely and thus easily covered by the noise of the measuring instrument.
Then, the gaps between the fitted Mooney-Rivlin fits, as shown in figure 3, are processed by Bayesian Change Point Detection to help determine the form of the compensation function for a given strain range.

Tensile tests and mooney-rivlin fits
After the compensation function was completed, tensile tests were performed in the temperature range of 281.5 to 340.2K, and raw data were corrected by the compensation function before fitting with the Mooney-Rivlin model.Specifically, the temperature was adjusted by pouring water into the tube, and then tension as a function of strain was measured for different water temperatures.By replacing the water out of the upper portion of the tube, the temperature may be efficiently changed using water prepared by mixing boiling water, roomtemperature water, and ice.Once the water is added to the system, the temperature is monitored with the electronic thermometer while the experimenter makes the measurements; several measurements can be made for different lengths before the temperature has rose or fell by as much as 0.5K, given the heat capacity of water.
For the tension measurement, since the maximum tension of the rubber band was merely above 0.25N, which is only slightly larger than the minimum scale value of a normal dynamometer, the normal dynamometer to measure the tension of the rubber band cannot be used.Also, the reading time of using a dynamometer is relatively slow, risking data to be subject to static fatigue effect.As an alternative, a high precision electronic scale with a minimum scale of 0.1g was used to measure the tension of the rubber band, which is the difference between the reading of mass times the gravitational acceleration in the slow and steady loading process.

Configurational entropy measurement
After the respective parameters C 10 and C 01 are obtained from the Mooney-Rivlin fits under different temperatures, the tensions under specific temperatures and strains may be determined.Next, the relationship between tension and temperature may be investigated upon by arranging tensile measurements to exhibit tension-temperature relation.Through regression analysis of the tension data under the corresponding strain within a strain amplitude 100%, the ( ) a best fit l values were determined by linear fits with a fitting step of 0.0001 cm, allowing for accuracy by approximating the integral as a Riemann sum, thereby obtaining the change in entropy.

Refined protocols for error reduction
For the two main aforementioned sources of error in tensile tests, static fatigue effect and the deviation of dynamic fatigue effect away from the linear-form segmented compensation function, respective methods are adopted to control these factors.
To reduce static fatigue effect.
(1) Although the thermal equilibrium tension should ideally be measured after sufficient relaxation time following elongation, full relaxation would allow for a static fatigue effect due to plastic flow after excessive duration compared with the experiment time [16].The video analysis method records the instantaneous pattern of continuous strain and corresponding tension change in a uniform uniaxial linear stretch, thereby removing rate-dependent plastic effect.
(2) To further mitigate the confusion between temperature effects and plasticity owing to static fatigue, temperature measurements were not performed in a monotonically increasing order.
To reduce deviations of linearity in compensation function.
(1) In the experiment, the strain amplitude, maximum loading cycles and temperature, were controlled to maintain the elastic deformation.Thus, the revised Mooney Rivlin fit may be applied to derive the linear form of the segmented compensation function [28].
(2) Preconditioning was applied to avoid rapid initial softening due to chain slippage in the first few cycles.The compensation function's coefficients for the latter four experiments (10th, 20th, 30th, and 40th) were simulated and then averaged to obtain a general compensation function.To prevent slippage, the first processed tensile data was from the 10th stretch.
(3) The experiment was conducted in a slow and constant stretching rate (d ε/dt) to minimize viscoelastic dynamics.
Moreover, several operational procedures were applied to reduce or recognize statistical uncertainties (1) To reduce the vertical parallax error, the video was taken at a distance from the acrylic tube, and the experimental object was placed in the center of the image.
(2) The Mooney-Rivlin fits were primarily fitted with high strain tensile data, as low strain data points are subject to random entropic behavior and the Mooney-Rivlin model assumes the passing from origin for every tensile test.
(3) Considering scale response time of about 2 seconds, the reading change threshold of 0.1g, and the maximum loading rate of 3.3 g s −1 , the overall measurement uncertainty of tension is ± 7g due to fluctuations in the measured values.

Preliminary experiment and compensation function
In this section, the tensile properties of four rubber bands are studied.After recording the tensile tests, Bayesian Change Point Detection is used to determine the form of compensation function at different ranges of strain, as shown in table 1.For the first form, by subtracting the integrated tension within each interval and dividing it by the interval length, the average damping value of tension is determined; and for the second form, the damping value is assumed to be in linear relation with strain, as shown on the graph's triangular shape in low strain regime.In both cases, the mean of each coefficient between every 10 cycles is taken in the general segmented compensation function.
While analyzing the tensile data, the entropic randomness and strain softening appears evident, serving as causes for the recession of compensation accuracy with increasing cross-sectional area.Several features related to the concept are reflected.
Figure 3 clearly exhibit a recession in dynamic fatigue effect from the 2 mm to the 1cm width rubber in low strain region.During the fortieth stretch of each rubber, the maximum elongation that has zero tensile strength in the Rubber band decreases monotonically, as shown in table 2.Moreover, within the strain amplitude of 200%, the enclosed proportion of the tension integration in the first tensile test exhibits a clear decreasing trend as the cross-sectional area increases.Such feature illustrates larger rubber band's resistivity against dynamic fatigue; from a mathematical perspective, this is the outcome of lower stress concentration for increasing elastomer volume.Specifically, the ratio of tension at specific strain tends to be smaller than that of the rubber band width, contrary to the estimation based on the Gaussian model, rendering larger rubbers less likely to reach undergo fatigue weakening.
Table 3 reveals the degree of compensation from the comparison between pre-post maximum and average deviation for the 4 rubber bands.While the compensation function may remove the dynamic fatigue effect for rubber bands with strongest tensile properties, it is shown to perform considerably better for weaker rubber bands in high strain regions.
One possible cause is the uncorrectable nature of the dynamic fatigue effect of 1cm and 5mm width rubbers' in relatively low strain regime, as their Mooney-Rivlin fit tends to intertwine below 50% strain.Another possible cause is the stronger elastic properties of the larger rubbers, rendering them less prone to strain softening, a key component in the molecular interactions described by the dynamic fatigue parameter; thus, compensation is less  4, the tensile data were plotted for a few different temperatures.The parameters of the (2 * 3) mm rubber band, including A 0 = 6mm 2 , were brought into the Mooney Rivlin model for fitting.The data in this figure were taken starting at the smallest strain (0%), moving to the longest (100%).The close proximity of the measured curves shows that the variation of the tension due to temperature (0∼0.5N) is small compared with its variation with strain(0∼2.5N).
The Mooney-Rivlin model provided and accurate fit for the 168 measured rubber band tension data points.The model achieved a correlation coefficient of above 0.996 for all isothermal conditions after compensation and a root mean square error (RMSE) 34.7% of that from Hooke's law.The Mooney Rivlin model demonstrates excellent agreement with the experimental data, providing a good mathematical description of the straintension behavior of the natural rubber fibers under the conditions tested.However, the Natural Rubber fiber used in this experiment reveals a discrepancy with the ideal Mooney Rivlin fit in low strain regimes, where a clear positive correlation between rubber band tension and temperature was not observed.This result is consistent with that of the preliminary experiment, which showed that the tension of elastomer is more variable at strains below 20%.

Results of configurational entropy change in isothermal stretching
After obtaining the temperature-dependent Mooney-Rivlin model parameters in table 4. Tension may be calculated for strains from 0% to 100% for all the given temperatures.Then, tension-temperature relation for a given strain may be graphed, and thereby ( ) a best fit l would be determined by measuring the slopes of the linear functions in figure 5.
In principle, / ( ) ¶ ¶ S l T will depend on temperature, but the linearity of the tension-temperature relation in figure 5 reveals roughly constant ( / ) ¶ ¶ S l T over the range of temperatures from 280K to 340K.Therefore, the configurational entropy changes up to a fixed strain can be determined by performing numerical integration with respect to length.( ) a best fit l is found with Vscode Python via the LMSE algorithm.The fitting step of a best fit was set to 0.0001 cm, and the values of entropy change under each elongation was then fitted by the summation of discrete a .

best fit
Aforementioned, a discrepancy appeared at low strains (0-2 cm, 13% strain), where the relationship between tension and temperature may in fact be negative due to thermoelastic inversion.By omitting several outlier data points where negative slope occurred, the change in entropy was fitted by numerically integrating infinitesimal changes and then compared to the fitted entropy changes from the Gaussian model (optimized by LMSE).Overall, figure 6 reveals that the Gaussian model performs relatively well in regions under 100% strain (correlation 0.9723) for predicting the entropy change of the rubber band.
As the change in entropy is observed to be independent of temperature during stretching, the maximum change in configurational entropy for the (160 * 2 * 3) mm rubber band is calculated via the Maxwell's relation.

Summary
The approximated Maxwell relation serves as a convenient tool for determining configurational entropy changes in Natural Rubber fibers under uniaxial tensile deformation.We developed, validated, and obtained encouraging results from the implementation of linear-form segmented compensation functions in calibrating tensile data at various temperatures.Post-compensation Mooney-Rivlin fits produced reliable tensile fits for the tension-strain properties of Natural Rubber fibers, evading noise coverage and dynamic fatigue effect's influence for even for high and low strain regimes; these extreme conditions are respectively subject to intensive strainsoftening and entropic randomness.Drawing the Mooney-Rivlin fits under various strains and temperatures, the values for tension-temperature coefficient ( ) a best fit l are obtained via regression analysis.A numerical integration of ( ) a best fit l consequently enables the determination of the configurational entropy change as a function of strain, serving as a simple quantification of deformation-induced entropy changes.
Although this paper investigates the stretching process, this method also has potential to be applied to the contraction process, after considering the hysteresis effect [16].Also, as the work done on rubber equals the change in Helmholtz free energy when the rubber is isothermally stretched (contracted), the change in internal energy for the isothermal stretch (contraction) may be found by the definition of the Helmholtz free energy F = U-TS (U being internal energy), after the change in entropy is determined by the depicted method.

best fit l 2 .
Data processing and the research-based implementation of the designed methods 2.1.Equipment and implementation 2.1.1.Equipment

Figure 2 .
Figure 2. Video analysis of instantaneous elongation of elastomer by Tracker (with corresponding tension reading of digital scale shown on the left of the image).

Figure 3 .
Figure 3. Depiction of the tensile data obtained from the 1st stretch and the 40th stretch for 160 * 2 * 2 mm, 160 * 2 * 3 mm, 160 * 2 * 5 mm, and 160 * 2 * 10 mm rubber bands at 330K in respective order, where the nth stretch is written as Tensile Experiment n, and its respective Mooney-Rivlin fit as Mooney Rivlin fit n.For readability, the 1st and the 40th stretch's tensile performance were graphed for comparison.

Figure 5 .
Figure 5. Tension versus temperature for each measured length.

Figure 6 .
Figure 6.Entropy change calculated using the Maxwell relation and Gaussian Model curve fit.

Table 3 .
Most deviated points, maximum deviations, and average deviations between the 1st and 40th loading cycles for 2 mm, 3 mm, 5 mm, 1 cm width rubber bands.

Table 1 .
General segmented compensation function for (2 * 2) mm, (2 * 3) mm, (2 * 5) mm, and (2 * 10) mm cross-sectional area rubber bands.applicable.Such property is reflected in the linearity of the wider rubber bands' Mooney-Rivlin fits, corresponding to the prediction of elasticity in Hooke's law, and the plateau for thinner rubber bands at high strains, indicating strain softening.

Table 4 .
Temperature-dependent parameters C 10 C 10 of the rubber band from K 340.2 to 281.5 K.