This is mainly due to the appearance of a rod like structure on neat PS which has improved the dispersion as well as provides a higher interface area that enhanced the UV-absorption efficiency of the PS matrix. This analysis is equally supported by the PALS study where the free volume was closely associated with the interfacial interaction between the filler and the PS matrix.
Journal of Science: Advanced Materials and Devices (2019) 413e419 Contents lists available at ScienceDirect Journal of Science: Advanced Materials and Devices journal homepage: www.elsevier.com/locate/jsamd Original Article Study of interfaces in polymer-metal oxide films and free-volume hole using low-energy positron lifetime measurements Aman Deep Acharya a, b, Bhawna Sarwan a, b, *, Ratnesh Sharma a, S.B Shrivastava a, Manoj Kumar Rathore c a b c Vikram University, Ujjain, 456010, MP, India Lovely Professional University, Jalandhar, Punjab, India M.P Council of Science and Technology, Bhopal, India a r t i c l e i n f o a b s t r a c t Article history: Received February 2019 Received in revised form 28 July 2019 Accepted 10 August 2019 Available online 16 August 2019 To reveal how the distribution of different nano fillers affect the UV-shielding efficiency of their polymerbased composites and to further develop a simple strategy to refrain the erection of the composites, we prepared ZnO doped polystyrene (PS/ZnO) and TiO2 doped polystyrene (PS/TiO2) films by the solution cast technique with different concentrations of ZnO and TiO2 (0.25%, 0.5%, 0.75% and 1%.) Contrary to the common observation, the better tunability for UV shielding efficiency was found in the case of TiO2 as compared to ZnO This is mainly due to the appearance of a rod like structure on neat PS which has improved the dispersion as well as provides a higher interface area that enhanced the UV-absorption efficiency of the PS matrix This analysis is equally supported by the PALS study where the free volume was closely associated with the interfacial interaction between the filler and the PS matrix These observations recommend that the better dispersion of filler particles leads a stronger interfacial interaction and enhances the UV-protection efficiency of the composite materials © 2019 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Keywords: Polystyrene thin films ZnO TiO2 Positron annihilation Free volume hole Interfacial interaction Solution cast method Introduction Previously, our group has successfully prepared TiO2/PS films with concentrations up to wt % by the solution cast method in the development of photo-protective polymeric materials for the protection against ultraviolet radiation Interestingly, the as-prepared thin films have shown a tremendous UV-shielding proficiency [1,2] These results suggested to pursue a further study on the relationships among the atomic free volume and the interfacial interaction between the filler particles and the PS matrix Contrary to the common observations where numerous approaches have been made for the development of ZnO doped composites as a better UV protective material, in our previous study TiO2 has expressed a better harmony for the efficient UV shielding which we will incorporate in the present work as a comparative study To analyze the imperfections produced at the early stage of the * Corresponding author Lovely Professional University, Jalandhar, Punjab, India E-mail addresses: acharyaphysics2011@gmail.com (A.D Acharya), sarbhawna@ gmail.com (B Sarwan) Peer review under responsibility of Vietnam National University, Hanoi process in engineering materials it is important to predict the weakness and failure of the material This work is consequential for the final understanding of the UV-shielding efficiency by comparing its results with those of a widely studied material as ZnO The very extensively studied inorganic materials ZnO and TiO2 with a wide band-gap energy of eV have been expansively used as inorganic UV absorbers due to their significant optical properties [3] Consequently, such polymer nanocomposites have been regarded as excellent candidates for UV shielding applications As a matter of fact, the extraordinary properties of the polymer nanocomposite include the dispersion of the nanoparticles in the matrix and the subsequent growth of enormous interfacial areas This complete dispersion allows the exploration of the available matrixeparticle interface and then the optimization of the organiceinorganic interaction which is accountable for the improved properties of the final material Nevertheless, there have been less reported on the research efforts in this area especially those dealing with the effect of the free volume hole and the interfacial interaction on the UV-shielding efficiency [4e6] Moreover, most of the works have adopted higher dopant concentrations to conquer a better UV-shielding efficiency of the films [4,7e10] https://doi.org/10.1016/j.jsamd.2019.08.003 2468-2179/© 2019 The Authors Publishing services by Elsevier B.V on behalf of Vietnam National University, Hanoi This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) 414 A.D Acharya et al / Journal of Science: Advanced Materials and Devices (2019) 413e419 whereas low content dosaging was constantly disregarded causing a shortage of understanding of the mechanism of the UV-shielding enhancement It would therefore be interesting to find out at which doping concentration the composite film starts to absorb the UV radiation to a significantly large extent In this contribution, we have prepared ZnO/PS and TiO2/PS films with dopant concentrations up to wt % by the solution cast method This is a green as well as simple method It is also useful in dissolving the polymers to have the material as finely divided as possible and to have each particle thoroughly wetted by the solvent Numerous techniques have been employed to analyze the impact of the interfacial interaction on the thermal conductivity, the thermal expansion, the viscoelastic properties and the density [11,12] In view of the applications the technique employed should have a strong influence on the estimation of the free volume The quantitative depiction of the free volume properties of polymers in the amorphous state can be accomplished by the use of the positron annihilation lifetime spectroscopy technique (PALS) This technique involves the insertion of positrons into the material and then recording the individual positron lifetimes until the annihilation with electrons of the sample takes place [11,12] Since the fraction of the positrons annihilates from the state of an orthopositronium (o-Ps) and the lifetime of the orthopositronium depends on the size of the free volume cavity where it is placed, hence, it can be employed to characterize the free volume size in amorphous polymers Following the above described study concern, the main motivation of the present work is to investigate what would be the effects of doping ZnO and TiO2 into PS on the atomic free volume, the interfacial contact among the filler particles and the PS matrix, and the UV-absorption efficiency of PS at low content of fillers by the positron annihilation lifetime spectroscopy Experimental ZnO/PS and TiO2/PS thin films with different concentrations viz 0.25%, 0.5%, 0.75% and 1% have been prepared by using the solution casting method The polystyrene was procured from the market which was in the granular form The PS solution was prepared in the dichloromethane, the requisite amount of semiconductors (ZnO or TiO2) was then added into the solution under rapid stirring for uniform dissolution The resultant was then poured on to a cleaned petri dish to cast the film and the solvent was subsequently allowed to evaporate gradually over a period of 12e24 h in a dry atmosphere The membrane was then physically peeled off from the surface The area of the cast surface, the material quantity and the density of the material can determine the thickness of the membrane We have prepared the polymer thin films of ~50 mm thickness For preparing the doped PS thin films, the dopant concentration was calculated from the following equation [1,13] Wwt%ị ẳ wf 100 wp þ wf where,wf and wp represent the weight of the dopant and the polymer, respectively X-ray diffraction patterns of ZnO/PS and TiO2/PS thin films were recorded on an X-ray diffractometer (Bruker D8 ADVANCE) with Cu-Ka radiation having the wavelength of 1.5418 Å in the range of 2q ¼ 200 - 700 Atomic force microscopy (AFM) measurements were carried out on a digital instrument of Nanoscope E with the Si3N4 100 mm cantilever and 0.58 N/m force constant The transmittance of the films has been measured with a UV-Vis Spectrophotometer (PerkinElmer Lambda 950) Measurements of the positron lifetime in ZnO/PS and TiO2/PS thin films have been done by using the slow e fast coincidence method Positron lifetime experiments are capable of distinguishing different kinds of defects As a spectroscopic technique the positron annihilation spectroscopy is a sensitive tool for the study of openvolume type defects which include vacancies, vacancy agglomerates, and dislocations A conventional fast-slow coincidence spectrometer was used to carry out the positron lifetime measurements having a resolution of 280 ps (FWHM) for Na22 source The Na22 source with strength of 20 mCi was deposited onto a nickel foil with the thickness of mg/cm2 and sandwiched between two similar pieces of the sample Nearly 105e106 counts were recorded in the PAL spectra for each sample Standard PATFIT program was used to analyze the positron lifetime spectra with their relative intensities Results and discussion 3.1 Structural and surface analysis The XRD patterns of the films are shown in Fig As it is perceived from the XRD patterns the pure PS film (Fig a,b) shows an amorphous polymeric structure and the diffraction peaks of PS not appearein the patterns The pattern of the PS thin films loaded with 0.5 wt% TiO2 and ZnO (Fig a,b), however, shows diffraction peaks of low intensity suggesting improved crystalinity of the PS At the higher filler content, the peak positions of the wt % sample is slightly shifted towards lower diffraction angles The most likely reason for this shift is the interaction between the filler particle and the polymer structure that leads to a rearrangement of the PS chains (See Fig 1a and b (Inset)) The increased intensity of the reflections from the diffracted planes with the higher amount of filler loadings suggests that a slowering of the crystallization rate arrised due to the enclosure of filler particles It can be concluded that a suitable, but not excessive, amount of dopant is responsible for observed good dispersion of the inorganic filler particles in the PS matrix To get a further insight, we extended our approach to another important analysis using the atomic force microscopy (AFM) of the doped polymer samples Figs and show the AFM images of thin sections of the ZnO/PS and TiO2/PS composite surfaces loaded with the dopant content of 0.25, 0.5, 0.75 and 1.0 wt % It can be observed from AFM images (Fig 2) of PS/ZnO that the grain size increases with the increase in the ZnO concentration upto 0.5 wt% (See Fig 2c) leading to the aggregation The addition of ZnO particles at about 0.75 wt % does not encourage the faster crystallization (See Fig 2d) In case of the 1.0 wt % sample (See Fig 2e), the efficiency of ZnO for enhancing the matrix crystallization get reduced due to the high particle density and obstructed the development of crystalline sections This illustrates that the small amount of ZnO particles i.e 0.5 wt% located in the PS matrix corresponds to the primary particles and the extent of the agglomeration was found to be quite negligible The PS matrix having 0.75 wt% and 1.0 wt% ZnO particles changed to large size aggregates where several primary ZnO nanocrystallites were gathered Furthermore, the entire morphology was deformed when 1.0 wt% ZnO was employed By comparing the AFM images, an obvious difference can be seen between the neat PS and the TiO2/PS film Fig 3a shows an AFM image of thin sections of the TiO2/PS thin film at the dopant content of 0.25 wt % It signifies that the TiO2 particles were setup in the form of aggregates of slackly linked paramount particles showing areas which are homogeneously implanted in the PS matrix After addition of 0.5 wt% TiO2 into the matrix some rods appeared (See Fig 3c) on the surface of the neat PS presuming that the formation of these rods mainly depends on the growth and nucleation conditions Moreover, a fractal type of aggregation of TiO2 particles has been observed in Fig 3d, such situation may arise due to the high concentration of nucleates that were formed by A.D Acharya et al / Journal of Science: Advanced Materials and Devices (2019) 413e419 415 Fig X-ray diffraction patterns: (a) ZnO/PS and (b) TiO2/PS thin films Fig AFM images of the ZnO/PS film adding the 0.75 wt% filler content Moreover, looking into Fig 3e, the nucleates randomly agglomerate in the continuous phase and cause the increase of the number of TiO2 particles, thus, making the interface area larger and the overlap of these led to opaquely appearing TiO2/PS films [9,13] This area is notably higher than that of the 0.5 wt% TiO2/PS films This observation suggests that the 0.75 wt% TiO2/PS films have large particle agglomerates, while the 0.5 wt % TiO2/PS films have an improved dispersion as well as a higher interface area and therefore exhibit a higher UV- absorption efficiency From this analysis, it may be inferred that to speed up the matrix crystallization and for altering the synthesized nanostructure morphology, a low concentration of dopant as such of 0.5 wt % is enough Herewith, the AFM results suggest that the inorganic semiconductor particles were well incorporated in the PS, which consequently modify significantly the morphology of the PS films 3.2 ZnO/PS and TiO2/PS UVevis shielding The transmittance characteristics of the pure PS, the ZnO/PS and TiO2/PSfilms are visualized in Fig It is found that almost 99% of the light was passed-on by the pure PS in the UVevis region of wavelengths from 300 to 700 nm As shown in Fig 4a, the maximum value of transmittance of the ZnO/PS films containing 416 A.D Acharya et al / Journal of Science: Advanced Materials and Devices (2019) 413e419 Fig AFM images of the TiO2/PS film Fig The transmittance spectra of (a) ZnO/PS and (b) TiO2/PS films 0.25 wt % ZnO was found as pretty as 95% By a careful consideration, it can be seen that the continuous inclusion of ZnO induces a systematic decline of the transmitted light, lowering the transmittance The transparency is also dependent on the dispersion/ aggregation of the nanoparticles into the polymer matrix The fractal distribution of discretely dispersed nanoparticles favors the optical transparent intensity loss of the transmitted light because the scattering abruptly rises with the particle size This causes a significant drop in the transparency of the films [10,14] In line with this, the gradual decrease in the visible-light transmission from 95 to 70% in the films containing 0.25e1 wt % ZnO was observed and highlighted by the shaded area in Fig 4a The thin films with wt % ZnO dopant shows a non-uniform distribution of the ZnO particles within the polymer matrix This could be endorsed by the AFM results (Fig 2) where no substantial ZnO agglomerations were found The obtained experimental results provide a visual illustration to the UV-shielding effect in ZnO/PS When ZnO/PS film is irradiated with the incident radiation, the visible light perfectly passes through the material as ZnO particles are apparent for the wavelengths greater than 375 nm while the UV-spectrum is obstructed depending on the ZnO concentration For the reason that the ZnO nanoparticles build a physical obstacle that the UV light cannot cross since they act as a protective network When the dopant concentration is further increased, the scattering mean free path gets decreased Due to this reason, the light traveled strongly inside the film with increased obstacle leading to the reduction in UV-light transmission to 63% with wt% ZnO concentration (see Fig 4a) The UVeVis transmittance of the TiO2/PS film is plotted in Fig 4b High transparency in both the visible and UV region is A.D Acharya et al / Journal of Science: Advanced Materials and Devices (2019) 413e419 Fig Plots of (a∙h∙y) 417 v/s photon energy (h∙y): (a) ZnO/PSand (b) TiO2/PS thin films observed in the pure PS film (see Fig 4b), which is not competent to filter out the UV radiations, whereas the addition of TiO2 content leads to the increase in the UV shielding efficiency due to the empty conduction band and the filled valence band However, the UV blocking consequence is seen in the films with TiO2 contents as low as 0.25 wt%, while the high transparency in the visible range is maintained The concentration of 0.5 wt % TiO2 could be assumed as the optimal one for the better UV shielding effect as evidenced by the graphical situation in the region bellow 355 nm, where more than 70% transparency is observed This evidences that the introduction of TiO2 particles into the PS matrix is compatible to increase the UV protecting proficiency of the PS film in the region from 300 to 355 nm The further increment of TiO2 (0.75 wt %) results in the opaque appearance with the increased absorption in both the visible and UV region In this state, the apparent nature of the material as a UV filter is decreased This behavior can be interpreted by the fact that the increased amount of TiO2 enhances the interface scattering causing the reduction in the transmittance This reduction might be ascribed to the growing cluster size [6] In addition, the cluster size of the film becomes more non-uniform, and irregular with the increasing TiO2 content up to wt% leading to the reduction in the transmittance as it is confirmed on the AFM images (Fig 3e) of the composite films Here, the shape of the PS latex is almost demolished and then totally vanished because of the interdiffusion process between the polymer chains From this result, it might apparently be easy to load the interstices of the thin PS template with a low concentration of dopant, but it is difficult to fill the interstices at a higher concentration So, the dopant content can be considered as a key parameter for the permeation of the PS templates [15] The band gap energies (Eg values) of the ZnO/PS and TiO2/PS films could be estimated from a plot of (ahn)2 vs the photon energy (hn) in Fig 5a,b Band gap values of 3.00, 2.47 and 2.61 eV were obtained for the pure PS, ZnO/PS and TiO2/PS films, respectively (for the optimum content, i.e 0.5%) However, two different mechanisms are accounted for the variation in the calculated optical band including: (1) The inclusion of a tiny amount of dopant produces charge transfer complexes in the host matrix which accelerate the electrical conductivity by providing additional charges which cause the reduction of the band gap [18,19]; (2) When the amount of dopants increases, the dopant molecules initiate to linking the gap between the localized states and thus lowering the potential barrier between them [16e22] 3.3 Positron annihilation lifetime studies The measurement for the positron annihilation lifetime studies (PALS) was carried out to examine the effect of TiO2 and ZnO on the microstructure of the composite films The positron lifetime spectra of the pure PS,ZnO/PS and TiO2/PS films, respectively, are presented in Fig They show a systematic decreasing trend of the lifetime This indicates a decrease in the longest lifetime The free volume size (Vf), and the o-Ps lifetime (t3) as a function of the TiO2 and ZnO content are shown in Fig 7aeb, respectively From Fig 7a, it can be observed that the t3 and Vf initially drop with the TiO2 incorporation upto 0.5 wt% In the range from 0.5 to 0.75 wt%, a slight increase in t3 and Vf is seen They finally decrease to the lowest value at higher doping, i.e at wt% Looking again into Fig 7a, there is a decrease in t3 and Vf with the increasing TiO2 concentration (0.25e0.5 wt %) indicating that the additional Fig Positron lifetime spectra: (a) ZnO/PSand (b) TiO2/PS thin films 418 A.D Acharya et al / Journal of Science: Advanced Materials and Devices (2019) 413e419 Fig The PALS results: o-Ps lifetime t3 and free volume size Vf of (a) TiO2 (b) ZnO amount of TiO2 slows down the o-Ps formation This can be explained by the fact that firstly the TiO2 particles fill up some of the free volume holes in the PS and so the values of t3 and Vf decrease Secondly, positrons may be annihilated from the TiO2 filler and there may be a lack of positrons which should be available to form the positronium in PS [12] On the other side, the increase of o-Ps at the dopant concentration of 0.75 wt% TiO2 suggests the formation of new positron trapping sites at the TiO2ePS interface As the filler concentration is increased to that corresponding to the wt% concentration, the TiO2 filler inhibits the o-Ps formation and the filler particles are scattered among the molecular chains of the PS and thus reducing the free volumes size leading to the decrease of the o-Ps lifetime in the PS Quite the reversal, a small but systematic increase in the free volume size and in the o-Ps lifetime has also been initially observed in the case of the low ZnO doping (i.e 0.25e0.5 wt%) (see Fig 7b) This is because of the development of new positron trapping sites at the ZnO/PS interface The highest values of t3 and Vf have been found for the 0.5 wt% ZnO/PS film, whereas when we have increased the ZnO concentration upto wt %, the values of t3 and Vf decrease showing that some of the free volume holes in the PS are filled up by the ZnO particles It is interesting to note that the interfacial interaction between the filler and the polymer matrix has caused a vital effect on the free volume size and the o-Ps lifetime This interaction dominates the delivery of phonons between the matrix and the fillers [23e25] mainly at 0.5 wt% ZnO concentration where both the free volume size and the o-Ps lifetime reach their maximum We recall the main fact that the film with a low dopant amount ZnO represents a high surface area, thus, providing more positron trapping sites which scatter the phonons at the interface [26e28] However, at the high ZnO concentrations, the ZnO agglomerates and due to this interfacial interaction, the induced disruption effect becomes limited, reducing the free volume size and the o-Ps lifetime 3.4 The correlation of PALS results and UV-shielding From the calculated results of the PALS and the morphological studies, the UV-shielding efficiency of the ZnO/PS and TiO2/PS films could be clearly understood From the PALS results of TiO2/PS, it is noticed that the free volume hole size and the o-Ps lifetime initially drop with the TiO2 incorporation It might be an evidence for the gradual formation of neutral aggregates at the initial level of filler concentrations which creates blockages and reduces the free volume holes, enhances the crystallization of the matrix as it was clearly confirmed by the AFM results This decrease in the free volume holes may also contribute to the increase in the UV shielding effect Furthermore, this factor reduces the ion and the segmental mobility through the unified matrix and hence, leads to the reduction of the free volume size At the high filler concentrations, a random distribution of filler particles might initiate the formation of free volume holes in the PS matrix This process of the free volume formation gradually dominates the creation of neutral aggregates which fairly agrees with the AFM results and confirms the transition from the crystalline state to the amorphous one at higher TiO2 concentrations where the regretable interaction between the loaded TiO2 particles and the PS matrix have slight limitations on the segmental mobility because of less contact area and so contributing to the increase in t3 and Vf as shown in Fig 7a An explanation based on the PALS results and detailed literature survey [27e29] implies that the o-Ps mainly annihilates in the interfacial regions In fact, there is some information indicating that the interfacial free volume is a vital factor for determinating the variation in the o-Ps annihilating lifetime because the interfaces have an excellent electronic density compared to the bulk phase It is worth noting that in the case of ZnO/PS, the initial increment in t3 with increasing ZnO concentration upto 0.5 wt% (see Fig 7b) suggests the formation of the free volume and amorphous phases in the blend matrix due to the sufficient separation between the filler particles at low filler concentrations Our tentative elucidation is that the dispersion of ZnO particles can cause the disorder of the molecular configuration leading to the morphological change in chains and thus increase in the free volume concentration and the o-Ps lifetime The above discussion corroborates that the better the dispersion of the lifetime the stronger will be the distraction of the molecular morphology On the other hand, as the ZnO content further increases, this distraction of ZnO weakens as being caused by the aggregation of ZnO leads to the decrease in the free volume concentration and the o-Ps lifetime The analysis also confirms that the free volume hole decreases with the increased filler concentration, because the filler limits the moving space of the molecular chains From the above explanation it can be clear that the calculated free volume hole size does not show a drastic variation with the ZnO content, but revealing a very little amount being adequate to accelerate the matrix crystallization which assures a less tunability for the UV radiation This could be probable because an excessive amount of ZnO can obstruct the formation of well crystallized regions and lead to the turbidity/translucency of the composite materials This illustrates that the particle size of the ZnO in the PS matrix corresponds to that of the primary particles and the extent of the agglomeration is moderately negligible, whereas in the case of TiO2 particles the reduction in the free volume hole size leads to the increase in the UV shielding effect It is evident that the more exposure of the PS to the UV causes the increase in the size of the free volume hole while the hole density remains unchanged The destruction of the PS matrix sets in when TiO2 is added leading to the formation of voids in the region of the TiO2 particles aggregates A.D Acharya et al / Journal of Science: Advanced Materials and Devices (2019) 413e419 This is attributed to the desorption and dispersion of the active oxygen species produced on TiO2 surface engraving the polymer matrix Furthermore, the rod shaped aggregation of the TiO2 nanoparticles (as shown in Fig 3c) acting as the nucleating agents leads to the increase in the UV shielding Moreover, doping with TiO2 results in a considerable increase in grain size due to the rod shape aggregation that leads to the reduction of the grain boundary scattering and this enhances the UV absorption and increases the visible transparency of the films To the best of our knowledge, the correlation between the UVabsorption behavior and the free volume hole was for the first time scrutinized Our experimental results clearly show that the PALS are useful to understand the UV-absorption efficiency of doped polymer film mixtures Conclusion The present research has explored the potential enhancement of polymer's UV-shielding properties The attention of our study has been focused on: (i) the analysis of properties of the atomic free volume defect, (ii) the filler-PS interfacial interaction and (iii) its impact on the UV-protecting adequacy of the films which have been investigated and examined by PALS Concussively, the shrink of free volume hole size due to the high filler concentrations is a supporting positron lifetime parameter The calculated value for the free volume hole size does not show any dramatic disparity with the high ZnO concentrations revealing the less tunability of the material for the UV radiation In the case of the TiO2 particles, however, the decrease in the free volume hole size has been observed because of the rod shaped aggregation of the TiO2 particles which act as nucleating agents contributing to the UV shielding efficiency of the PS The results as obtained suggest that TiO2 and ZnO acting as active fillers for PS can be used for improving the tremendous photo-protective shielding quality of the polymeric materials to be applied in ultraviolet radiation protection Howbeit, due to the surface free energy of the nanocrystals, ZnO particles tend to aggregate making them obscured to attain a homogeneous dispersal and that results in opaque composite films Therefore, the main efforts should be focused on the nanocrystals without aggregation in the PS Acknowledgements The authors are grateful to Dr Y.K Vijay (Prof.) at University of Rajasthan, Jaipur for providing the experimental facilities to study the positron lifetime and Dr Balram Tripathi for helping in the analysis of 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distinguishing different kinds of defects As a spectroscopic technique the positron. .. Looking again into Fig 7a, there is a decrease in t3 and Vf with the increasing TiO2 concentration (0.25e0.5 wt %) indicating that the additional Fig Positron lifetime spectra: (a) ZnO/PSand... chains of the PS and thus reducing the free volumes size leading to the decrease of the o-Ps lifetime in the PS Quite the reversal, a small but systematic increase in the free volume size and in