Polymers

2024

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25 pages, 6909 KiB

Open AccessArticle

Calibration of Thermal Viscoelastic Material Models for the Dynamic Responses of PVB and SG Interlayer Materials

by Jon Knight, Hani Salim, Hesham Elemam and Ahmed Elbelbisi

Polymers 2024, 16(13), 1870; https://doi.org/10.3390/polym16131870 - 30 Jun 2024

Viewed by 820

Abstract

Laminated glass interlayer materials polyvinyl butyral (PVB) and SentryGlas^® (SG, kuraray, Houstan, TX, USA) exhibit thermal viscoelastic behavior under dynamic tensile loading. Significant temperature and strain rate effects on the behavior of these interlayer materials pose a challenge for accurately modeling the [...] Read more.

Laminated glass interlayer materials polyvinyl butyral (PVB) and SentryGlas^® (SG, kuraray, Houstan, TX, USA) exhibit thermal viscoelastic behavior under dynamic tensile loading. Significant temperature and strain rate effects on the behavior of these interlayer materials pose a challenge for accurately modeling the dynamic response of laminated glass. Many researchers have simplified their approaches by modeling the response of the interlayer material using a bilinear approximation or established hyperelastic models. However, temperature and strain rate effects can be captured using the three-network viscoplastic (TNV) model. Therefore, the objective of this study is to calibrate material models for the thermal viscoelastic dynamic responses of PVB and SG interlayer materials. Uniaxial tensile tests were performed at strain rates of 2, 20, and 45 s⁻¹ and temperatures of 0, 23, and 60 °C, and material models were calibrated using the experimental data. Finite element analysis using the calibrated material models successfully predicted the dynamic responses of PVB and SG under the experimental test conditions within a 10% error margin. This suggests that the calibrated models using the TNV model represent significant improvements over existing approaches to modeling the dynamic response of laminated glass. Similar procedures can be applied to other thermoplastics, laying the groundwork for establishing a standard calibration guide. Full article

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2022

Jump to: 2024, 2021, 2020

9 pages, 1234 KiB

Open AccessArticle

Analyzing the Mechanical Properties of Free-Standing PACA Thin Films Using Microindentation Technique

by Osamah Altabal and Christian Wischke

Polymers 2022, 14(22), 4863; https://doi.org/10.3390/polym14224863 - 11 Nov 2022

Cited by 2 | Viewed by 1752

Abstract

Assessing the mechanical properties of materials is of fundamental relevance for their rational usage, but can be challenging with standard tensile testing for highly brittle polymers used, e.g., as coatings. Here, a procedure for the mechanical analysis of free-standing poly(alkyl cyanoacrylate) (PACA) films [...] Read more.

Assessing the mechanical properties of materials is of fundamental relevance for their rational usage, but can be challenging with standard tensile testing for highly brittle polymers used, e.g., as coatings. Here, a procedure for the mechanical analysis of free-standing poly(alkyl cyanoacrylate) (PACA) films using microindentation has been explored. Rigid and transparent films from PACA with various side chain compositions were formed on top of square polymer frames by in situ polymerization. Under microscopic control, the free-standing films were analyzed using a microelectromechanical sensing system. By this procedure, decreasing Young’s moduli E for increasing PACA side chain length and flexibility were determined with strain at break ε_B between 0.36% for poly(ethyl cyanoacrylate) and 4.6% for poly(methoxyethyl cyanoacrylate). Based on this successful application, the applied methodology may be relevant for characterizing various coating materials, which are otherwise hard to form as thin free-standing films, and using the data, e.g., in computationally assisted design and evaluation of hybrid material devices. Full article

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14 pages, 4472 KiB

Open AccessArticle

Quantitative Characterization of the Anisotropy of the Stress-Optical Properties of Polyethylene Terephthalate Films Based on the Photoelastic Method

by Quanyan He, Miaojing Wang, Yitao Du, Qinghua Qin and Wei Qiu

Polymers 2022, 14(16), 3257; https://doi.org/10.3390/polym14163257 - 10 Aug 2022

Cited by 4 | Viewed by 2605

Abstract

Polyethylene terephthalate (PET) is one of the most commonly used substrate materials in the field of flexible electronics, and its stress-induced birefringence often has a detrimental effect on the optical properties of the device. Therefore, a deep and systematic understanding of the stress-optical [...] Read more.

Polyethylene terephthalate (PET) is one of the most commonly used substrate materials in the field of flexible electronics, and its stress-induced birefringence often has a detrimental effect on the optical properties of the device. Therefore, a deep and systematic understanding of the stress-optical properties of PET films is crucial for device design and manufacture. The photoelastic method is a direct optical measurement technique based on the stress-induced birefringence effect of materials, which has the advantages of being nondestructive and noncontact. In this work, the photoelastic method was used to quantitatively characterize the anisotropy of the stress-optical properties of PET films under the uniaxial stress state. First, a self-built reflection-transmission coaxial bidirectional photoelasticity measurement system was developed by means of a combination of transmission and reflection photoelasticity. Then, the stress-optical coefficients and isoclinic angles of PET films with different stretching angles were measured. Finally, the linear combinations of the photoelastic tensor components and refractive-index-related parameters were determined by fitting the analytical relationship between the stress-optical coefficients and isoclinic angles. Full article

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42 pages, 8439 KiB

Open AccessReview

Shape-Memory Materials via Electrospinning: A Review

by Valentina Salaris, Adrián Leonés, Daniel Lopez, José Maria Kenny and Laura Peponi

Polymers 2022, 14(5), 995; https://doi.org/10.3390/polym14050995 - 28 Feb 2022

Cited by 25 | Viewed by 5111

Abstract

This review aims to point out the importance of the synergic effects of two relevant and appealing polymeric issues: electrospun fibers and shape-memory properties. The attention is focused specifically on the design and processing of electrospun polymeric fibers with shape-memory capabilities and their [...] Read more.

This review aims to point out the importance of the synergic effects of two relevant and appealing polymeric issues: electrospun fibers and shape-memory properties. The attention is focused specifically on the design and processing of electrospun polymeric fibers with shape-memory capabilities and their potential application fields. It is shown that this field needs to be explored more from both scientific and industrial points of view; however, very promising results have been obtained up to now in the biomedical field and also as sensors and actuators and in electronics. Full article

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20 pages, 5340 KiB

Open AccessArticle

Airbrushed Polysulfone (PSF)/Hydroxyapatite (HA) Nanocomposites: Effect of the Presence of Nanoparticles on Mechanical Behavior

by Monireh Moradienayat, Dania Olmos and Javier González-Benito

Polymers 2022, 14(4), 753; https://doi.org/10.3390/polym14040753 - 15 Feb 2022

Cited by 6 | Viewed by 2132

Abstract

Nanocomposite films of polysulfone (PSF)—hydroxyapatite (HA) were prepared with a commercial airbrush. Structural, thermal, and mechanical characterization allows obtaining new information to understand the role of the nanofiller–polymer matrix interphase in the final performance of these materials in relation to its possible applications [...] Read more.

Nanocomposite films of polysulfone (PSF)—hydroxyapatite (HA) were prepared with a commercial airbrush. Structural, thermal, and mechanical characterization allows obtaining new information to understand the role of the nanofiller–polymer matrix interphase in the final performance of these materials in relation to its possible applications in the restoration of bones. Fourier-transform infrared spectroscopy shows that there are hardly any structural changes in the polymer when adding HA particles. From thermal analysis (differential scanning calorimetry and thermogravimetry), it can be highlighted that the presence of HA does not significantly affect the glass transition temperature of the PSF but decelerates its thermal degradation. All this information points out that any change in the PSF performance because of the addition of HA particles cannot be due to specific interactions between the filler and the polymer. Results obtained from uniaxial tensile tests indicate that the addition of small amounts of HA particles (1% wt) leads to elastic moduli higher than the upper bound predicted by the rule of mixtures suggesting there must be a high contribution of the interphase. A simple model of the nanocomposite is proposed for which three contributions must be considered, particles, interphase and matrix, in such a way that interphases arising from different particles can interact by combining with each other thus leading to a decrease in its global contribution when the amount of particles is high enough. The mechanical behavior can be explained considering a balance between the contribution of the interphase and the number of particles. Finally, a particular mechanism is proposed to explain why in certain nanocomposites relatively high concentrations of nanoparticles may substantially increase the strain to failure. Full article

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2021

Jump to: 2024, 2022, 2020

19 pages, 3804 KiB

Open AccessArticle

Molecular Dynamics Investigation of the Thermo-Mechanical Properties of the Moisture Invaded and Cross-Linked Epoxy System

by Can Sheng, Gai Wu, Xiang Sun and Sheng Liu

Polymers 2022, 14(1), 103; https://doi.org/10.3390/polym14010103 - 28 Dec 2021

Cited by 16 | Viewed by 2401

Abstract

In spite of a high market share of plastic IC packaging, there are still reliability issues, especially for the effects of moisture. The mechanism between moisture and epoxy polymer is still obscure. A multi-step cross-linking approach was used to mimic the cross-linking process [...] Read more.

In spite of a high market share of plastic IC packaging, there are still reliability issues, especially for the effects of moisture. The mechanism between moisture and epoxy polymer is still obscure. A multi-step cross-linking approach was used to mimic the cross-linking process between the DGEBA resin and JEFFAMINE^®-D230 agent. Based on the molecular dynamics method, the thermo-mechanical properties and microstructure of epoxy polymer were analyzed. In this paper, the degree of cross-linking ranged from 0% to 85.4% and the moisture concentration ranged from 0 wt.% to 12 wt.%. The hydrogen bonds were investigated in the moisture invaded epoxy polymer. Although most of the hydrogen bonds were related to water molecules, the hydrogen bonds between the inside of epoxy polymer were reduced only a little as the concentration of moisture increased. The diffusion coefficient of the water molecules was found to increase with the increase of moisture concentration. When the moisture concentration was larger than 12 wt.% or smaller than 1.6 wt.%, the diffusion coefficient was less affected by the epoxy polymer. In addition, the free volume and the thermal conductivity of the epoxy polymer were considered. It was found that the moisture could increase the thermal conductivity from 0.24 to 0.31 W/m/K, identifying a coupling relationship between moisture and thermal properties. Finally, the mechanical properties of epoxy polymer were analyzed by uniaxial tensile simulation. The COMPASS and DREIDING force fields were used during the uniaxial tensile simulation. A better result was achieved from the DREIDING force field compared with the experiment. The degree of cross-linking was positively correlated with mechanical properties. For the system with the largest degree of cross-linking of 85.4%, the Young’s modulus was 2.134 ± 0.522 GPa and the yield strength was 0.081 ± 0.01 GPa. There were both plasticizing and anti-plasticizing effects when the water molecules entered the epoxy polymer. Both the Young’s moduli and yield strength varied in a large range from 1.38 to 2.344 GPa and from 0.062 to 0.128 GPa, respectively. Full article

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17 pages, 5329 KiB

Open AccessArticle

Research on Compressive and Flexural Properties of Coal Gangue-Slag Geopolymer under Wetting-Drying Cycles and Analysis of Micro-Mechanism

by Xiaoyun Yang, Yan Zhang, Zhuhan Li and Minglei Wang

Polymers 2021, 13(23), 4160; https://doi.org/10.3390/polym13234160 - 28 Nov 2021

Cited by 2 | Viewed by 1848

Abstract

Coal gangue-slag geopolymer is a kind of environment-friendly material with excellent engineering performance and is formed from coal gangue and slag after excitation by an alkaline activator. In this study, three kinds of coal gangue-slag geopolymer were activated by different activators, and the [...] Read more.

Coal gangue-slag geopolymer is a kind of environment-friendly material with excellent engineering performance and is formed from coal gangue and slag after excitation by an alkaline activator. In this study, three kinds of coal gangue-slag geopolymer were activated by different activators, and the compressive and flexural strengths of water and sulphate solutions in the wetting-drying (W-D) cycles were compared. The microscopic mechanism was analyzed by the XRD, the FTIR and the SEM. The following conclusions are drawn: The influence of W-D cycles on flexural strength was greater than compressive strength. The water migration and the recombination of geopolymers lead to the change of colour, as well as the reduction of flexural strength and compressive strength of geopolymers. The SH geopolymer had excellent anti-erosion ability in terms of flexural strength, and the reason for this was the recombination and polymerization reaction of geopolymer being weaker than the SS and the SSG. The corrosion resistance of the SS was reflected in the compressive strength, because its geopolymerization reaction was fierce, which produced more Na-rich C–N–A–S–H, N–A–S–H and C–A–S–H gels. Therefore, the compressive strength could still reach more than 39 MPa after 150 cycles. Sulfate solution could effectively control the reduction of compressive strength of the SH and the SS geopolymers during W-D cycles. The SSG had the worst corrosion resistance. Full article

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17 pages, 3128 KiB

Open AccessArticle

Nanoindentation Reveals Crosslinking Behavior of Solar Encapsulants—The Methodological Advantages over Bulk Methods

by Djamel Eddine Mansour, Christoph Herzog, Petra Christöfl, Luciana Pitta Bauermann, Gernot Oreski, Andreas Schuler, Daniel Philipp and Paul Gebhardt

Polymers 2021, 13(19), 3328; https://doi.org/10.3390/polym13193328 - 29 Sep 2021

Cited by 1 | Viewed by 2200

Abstract

The power degradation and failure of photovoltaic (PV) modules can be caused by changes in the mechanical properties of the polymeric components during the module lifetime. This paper introduces instrumented nanoindentation as a method to investigate the mechanical properties of module materials such [...] Read more.

The power degradation and failure of photovoltaic (PV) modules can be caused by changes in the mechanical properties of the polymeric components during the module lifetime. This paper introduces instrumented nanoindentation as a method to investigate the mechanical properties of module materials such as polymeric encapsulants. To this end, nanoindentation tests were carried out on ethylene vinyl acetate (EVA) surfaces, which have been separated from the glass panel. Two types of time-dependent indentation cycle modes, the time domain (creep mode) and frequency domain (dynamic mode) were performed to determine the viscoelastic behavior. For each mode, a corresponding model was applied to calculate the main mechanical properties. The general capability of nanoindentation as cross-linking determination method is investigated with the methodological advantages over bulk mechanical characterization methods. A large number of Glass/EVA/Backsheet laminates were built using different lamination conditions resulting in different degrees of curing. Both indentation modes indicate good modulus sensitivity for following the EVA crosslinking in its early stages but could not reliably differentiate between samples with higher EVA branching. Additional dynamic mechanical analysis (DMA) characterization was used as an established method to validate the indentation measurements. Both nanoindentation and DMA tensile mode produce similar quantitative viscoelastic responses, in the form of the damping factor parameter, demonstrated for three different frequencies at room temperature. A statistical study of the data reveals the advantages for the investigation of multilayer PV laminates by using nanoindenation as a surface method while also being applicable to field aged modules. Full article

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20 pages, 6171 KiB

Open AccessArticle

Comparative Study of Machine Learning Approaches for Predicting Creep Behavior of Polyurethane Elastomer

by Chunhao Yang, Wuning Ma, Jianlin Zhong and Zhendong Zhang

Polymers 2021, 13(11), 1768; https://doi.org/10.3390/polym13111768 - 28 May 2021

Cited by 11 | Viewed by 3334

Abstract

The long-term mechanical properties of viscoelastic polymers are among their most important aspects. In the present research, a machine learning approach was proposed for creep properties’ prediction of polyurethane elastomer considering the effect of creep time, creep temperature, creep stress and the hardness [...] Read more.

The long-term mechanical properties of viscoelastic polymers are among their most important aspects. In the present research, a machine learning approach was proposed for creep properties’ prediction of polyurethane elastomer considering the effect of creep time, creep temperature, creep stress and the hardness of the material. The approaches are based on multilayer perceptron network, random forest and support vector machine regression, respectively. While the genetic algorithm and k-fold cross-validation were used to tune the hyper-parameters. The results showed that the three models all proposed excellent fitting ability for the training set. Moreover, the three models had different prediction capabilities for the testing set by focusing on various changing factors. The correlation coefficient values between the predicted and experimental strains were larger than 0.913 (mostly larger than 0.998) on the testing set when choosing the reasonable model. Full article

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2020

Jump to: 2024, 2022, 2021

17 pages, 4700 KiB

Open AccessArticle

A Combined Exponential-Power-Law Method for Interconversion between Viscoelastic Functions of Polymers and Polymer-Based Materials

by Vitor Dacol, Elsa Caetano and João R. Correia

Polymers 2020, 12(12), 3001; https://doi.org/10.3390/polym12123001 - 16 Dec 2020

Cited by 1 | Viewed by 2204

Abstract

Understanding and modeling the viscoelastic behavior of polymers and polymer-based materials for a wide range of quasistatic and high strain rates is of great interest for applications in which they are subjected to mechanical loads over a long time of operation, such as [...] Read more.

Understanding and modeling the viscoelastic behavior of polymers and polymer-based materials for a wide range of quasistatic and high strain rates is of great interest for applications in which they are subjected to mechanical loads over a long time of operation, such as the self-weight or other static loads. The creep compliance and relaxation functions used in the characterization of the mechanical response of linear viscoelastic solids are traditionally determined by conducting two separate experiments—creep tests and relaxation tests. This paper first reviews the steps involved in conducting the interconversion between creep compliance and relaxation modulus in the time domain, illustrating that the relaxation modulus can be obtained from the creep compliance. This enables the determination of the relaxation modulus from the results of creep tests, which can be easily performed in pneumatic equipment or simple compression devices and are less costly than direct relaxation tests. Some existing methods of interconversion between the creep compliance and the relaxation modulus for linear viscoelastic materials are also presented. Then, a new approximate interconversion scheme is introduced using a convenient Laplace transform and an approximated Gamma function to convert the measured creep compliance to the relaxation modulus. To demonstrate the accuracy of the fittings obtained with the method proposed, as well as its ease of implementation and general applicability, different experimental data from the literature are used. Full article

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10 pages, 1834 KiB

Open AccessArticle

A Magneto-Hyperelastic Model for Silicone Rubber-Based Isotropic Magnetorheological Elastomer under Quasi-Static Compressive Loading

by Yanliang Qiao, Jiangtao Zhang, Mei Zhang, Lisheng Liu and Pengcheng Zhai

Polymers 2020, 12(11), 2435; https://doi.org/10.3390/polym12112435 - 22 Oct 2020

Cited by 9 | Viewed by 2568

Abstract

A new magneto-hyperelastic model was developed to describe the quasi-static compression behavior of silicone rubber-based isotropic magnetorheological elastomer (MRE) in this work. The magnetization property of MRE was characterized by a vibrating sample magnetometer (VSM), and the quasi-static compression property under different magnetic [...] Read more.

A new magneto-hyperelastic model was developed to describe the quasi-static compression behavior of silicone rubber-based isotropic magnetorheological elastomer (MRE) in this work. The magnetization property of MRE was characterized by a vibrating sample magnetometer (VSM), and the quasi-static compression property under different magnetic fields was tested by using a universal testing machine equipped with a magnetic field accessory. Experimental results suggested that the stiffness of the isotropic MRE increased with the magnetic flux density within the tested range. Based on experimental results, a new magneto-hyperelastic model was established by coupling the Ogden hyperelastic model, the magnetization model and the magneto-induced modulus model based on a magnetic dipole theory. The results show that the proposed new model can accurately predict the quasi-static compression property of the isotropic MRE under the tested magnetic flux density and strain ranges using only three model parameters. Full article

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18 pages, 3094 KiB

Open AccessArticle

Mechanical and Antimicrobial Polyethylene Composites with CaO Nanoparticles

by Cristián Silva, Felipe Bobillier, Daniel Canales, Francesca Antonella Sepúlveda, Alejandro Cament, Nicolás Amigo, Lina M. Rivas, María T. Ulloa, Pablo Reyes, J. Andrés Ortiz, Tatiana Gómez, Carlos Loyo and Paula A. Zapata

Polymers 2020, 12(9), 2132; https://doi.org/10.3390/polym12092132 - 18 Sep 2020

Cited by 29 | Viewed by 4490

Abstract

Low-density polyethylene composites containing different sizes of calcium oxide (CaO) nanoparticles were obtained by melt mixing. The CaO nanoparticles were synthesized by either the sol-gel or sonication methods, obtaining two different sizes: ca. 55 nm and 25 nm. These nanoparticles were used either [...] Read more.

Low-density polyethylene composites containing different sizes of calcium oxide (CaO) nanoparticles were obtained by melt mixing. The CaO nanoparticles were synthesized by either the sol-gel or sonication methods, obtaining two different sizes: ca. 55 nm and 25 nm. These nanoparticles were used either as-synthesized or were modified organically on the surface with oleic acid (Mod-CaO), at concentrations of 3, 5, and 10 wt% in the polymer. The Mod-CaO nanoparticles of 25 nm can act as nucleating agents, increasing the polymer’s crystallinity. The Young’s Modulus increased with the Mod-CaO nanoparticles, rendering higher reinforcement effects with an increase as high as 36%. The reduction in Escherichia coli bacteria in the nanocomposites increased with the amount of CaO nanoparticles, the size reduction, and the surface modification. The highest antimicrobial behavior was found in the composites with a Mod-CaO of 25 nm, presenting a reduction of 99.99%. This strong antimicrobial effect can be associated with the release of the Ca²⁺ from the composites, as studied for the composite with 10 wt% nanoparticles. The ion release was dependent on the size of the nanoparticles and their surface modification. These findings show that CaO nanoparticles are an excellent alternative as an antimicrobial filler in polymer nanocomposites to be applied for food packaging or medical devices. Full article

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15 pages, 3632 KiB

Open AccessArticle

Influence of the Degree of Cure in the Bulk Properties of Graphite Nanoplatelets Nanocomposites Printed via Stereolithography

by Alberto S. De León and Sergio I. Molina

Polymers 2020, 12(5), 1103; https://doi.org/10.3390/polym12051103 - 12 May 2020

Cited by 23 | Viewed by 3347

Abstract

In this work, we report on the fabrication via stereolithography (SLA) of acrylic-based nanocomposites using graphite nanoplatelets (GNPs) as an additive. GNPs are able to absorb UV–Vis radiation, thus blocking partial or totally the light path of the SLA laser. Based on this, [...] Read more.

In this work, we report on the fabrication via stereolithography (SLA) of acrylic-based nanocomposites using graphite nanoplatelets (GNPs) as an additive. GNPs are able to absorb UV–Vis radiation, thus blocking partial or totally the light path of the SLA laser. Based on this, we identified a range of GNP concentrations below 2.5 wt %, where nanocomposites can be successfully printed. We show that, even though GNP is well-dispersed along the polymeric matrix, nanocomposites presented lower degrees of cure and therefore worse mechanical properties when compared with pristine resin. However, a post-processing at 60 °C with UV light for 1 h eliminates this difference in the degree of cure, reaching values above 90% in all cases. In these conditions, the tensile strength is enhanced for 0.5 wt % GNP nanocomposites, while the stiffness is increased for 0.5–1.0 wt % GNP nanocomposites. Finally, we also demonstrate that 2.5 wt % GNP nanocomposites possess characteristic properties of semiconductors, which allows them to be used as electrostatic dispersion materials. Full article

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13 pages, 2700 KiB

Open AccessArticle

Poly(lactic Acid)–Biochar Biocomposites: Effect of Processing and Filler Content on Rheological, Thermal, and Mechanical Properties

by Rossella Arrigo, Mattia Bartoli and Giulio Malucelli

Polymers 2020, 12(4), 892; https://doi.org/10.3390/polym12040892 - 12 Apr 2020

Cited by 93 | Viewed by 5912

Abstract

Biocomposites based on poly(lactic acid) (PLA) and biochar (BC) particles derived from spent ground coffee were prepared using two different processing routes, namely melt mixing and solvent casting. The formulated biocomposites were characterized through rheological, thermal, and mechanical analyses, aiming at evaluating the [...] Read more.

Biocomposites based on poly(lactic acid) (PLA) and biochar (BC) particles derived from spent ground coffee were prepared using two different processing routes, namely melt mixing and solvent casting. The formulated biocomposites were characterized through rheological, thermal, and mechanical analyses, aiming at evaluating the effects of the filler content and of the processing method on their final properties. The rheological characterization demonstrated the effectiveness of both exploited strategies in achieving a good level of filler dispersion within the matrix, notwithstanding the occurrence of a remarkable decrease of the PLA molar mass during the processing at high temperature. Nevertheless, significant alterations of the PLA rheological behavior were observed in the composites obtained by melt mixing. Differential scanning calorimetry (DSC) measurements indicated a remarkable influence of the processing method on the thermal behavior of biocomposites. More specifically, melt mixing caused the appearance of two melting peaks, though the structure of the materials remained almost amorphous; conversely, a significant increase of the crystalline phase content was observed for solvent cast biocomposites containing low amounts of filler that acted as nucleating agents. Finally, thermogravimetric analyses suggested a catalytic effect of BC particles on the degradation of PLA; its biocomposites showed decreased thermal stability as compared with the neat PLA matrix. Full article

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