Electronic Supplementry Mteril (ESI) for Physicl Chemistry Chemicl Physics. This journl is the Owner Societies 2017 Supporting Informtion Mechnism of Roughness-induced CO 2 Microbubble Nucletion in Polypropylene Foming Linyn Wng,c, Wei Zhng c, Xingdong Wng,*, Jinguo Mi b, Jingjun M c, nd Zhongjie Du b,* School of Mterils nd Mechnicl Engineering, Beijing Technology nd Business University, Beijing, Chin b The Key Lbortory of Crbon Fiber nd Functionl Polymers, Ministry of Eduction, Beijing University of Chemicl Technology, Beijing, Chin c Institute of Technology in Bohi Cmpus, Agriculturl University of Hebei, Chin 1. Clcultion of Helmholtz free energy The fundmentl mesure theory is commonly used to describe the contribution of hrd-sphere repulsion, 1 which is expressed s hs { ( )} hs F kbt n d r r r hs where the detil informtion n cn be seen elsewhere. 2 r (S1) With the weighted density pproximtion, the ttrctive prt cn be expressed s 3 tt F [{ ( r)}] k T ( r){ F [ ( r)] F [ ( r)]} dr, (S2) B 1 2 * Corresponding uthors: wngxid@th.btbu.edu.cn; duzj@mil.buct.edu.cn
in which the expressions of F [ 1 ( )] r nd 2 ( )] r re given by the first-order men sphericl pproximtion expnsion z R 1 z R 1, 1, F1 [ ( r)] 2 ( r) x x k G ( z ) e 1, 0, 1, 2 z 1, 1 z R k G z e x x R I z R 2, 2, 3 ( ) 8 ( ) 2 2, 0, 2, r, z 2, 3 8 ( r) x x g ( R ) R I 0,,1 3 1,,1 z R z R 1, 2, F2 [ ( r)] ( r) xx k G ( z ) e k G ( z ) e 1, 1, 1, 2, 2, 2, 4 ( r) x x g ( R ) R I (S3) (S4) The weighted density ( r) is defined s tt ( r) ( r ) ( r r )dr, (S5) where tt ( ) is the weight function tt tt tt tt ij ( r) u r / u d, nd is the LJ ttrctive r r u ( ) r r interction potentil between ny two sites of species nd. The free energy contribution from the moleculr connectivity re bsed on the ssociting theory 4 Nsite chin site X ( r ) 1 1 F { ( r)} dr1 ( r1 ) ln X ( r1 ), (S6) 1 2 2 where mens the set of ll the ssociting points on site, X corresponds to the frction of site which re not bonded t point, nd N indictes the totl site number of polymer chin. The first site summtion of Nsite is over ll the site, nd the second is over ll the ssocition points on site. The free energy functionl due to the contribution of chin conformtion is constructed to ccount for the polymer configurtionlly entropy F conf 1 { ( r)} dr d r ( r) ( r) ( r- r ) 2 conf c (S7)
conf semiflexible flexible with the pproximtion c ( r) c ( r) c ( r). Here c semiflexible ( r) nd c flexible ( r) stnd for the direct correltion functions of semiflexible nd flexible PP chins. They re clculted from the polymer reference interction site model integrl eqution 5 r r r r r r h( r) dr dr ( ) c( )[ ( r ) h( r )], (S8) where h( r ) is the totl correltion function, ( r) the intrmoleculr correltion function. ( r) for flexible or semiflexible chins re represented by the Koym model. 6,7 The Kovlenko Hirt pproximtion 8 is dopted to solve the eqution. 2. The externl potentil of polymer fluid nd nnoprticle fluid The interction between fluid molecule site nd polymer site p is given by the conventionl Lenrd Jones (LJ) potentil U r 4 p p p 12 6 p p rp rp (S9) nd r p denotes to the loctions of the intercting sites. The LJ prmeters for unlike sites re clculted using the Lorentz Berthelot mixing rules. The interction between fluid molecule site nd F-POSS prticle n cn be clculted by 9 D n n r exp U n n r (S10) Dn Where is the mixed dimeter of F-POSS prticle nd fluid molecule, the sptil rnge prmeter is mteril dependent, nd its vlue is 0.5, r n represents the center distnce between F-POSS prticle nd fluid molecule site. 3. Construction of composite surfces
The crystlline PP structure is directly constructed using the Mteril Studio. The morphous PP structure is constructed using the moleculr dynmics (MD) simultion. During the simultion of morphous PP surfce, PP chins re plced in lrge periodic box to construct the initil configurtion. NPT ensemble is employed to compress the systems of low density to the desired density. The system is equilibrted for t lest 6 ns with NVT simultions under one br pressure. Such equilibrted structure is used s initil configurtion for further production run of (10-20) 10 6 MD steps by NPT simultion. During ll simultion runs, time step of 1.0 fs is used with multiple time step integrtor. 10 Isotherml clcultions re crried out with Nose-Hoover thermostt using coupling frequency of 0.02fs -1 nd the pressure is controlled by Nose-Hoover brostt. 11 The bond lengths re constrined using the SHAKE lgorithm. 12 The morphous or crystlline PP surfces cn be represented by slice of the constructed system. Afterwrds, different nnocomposite surfces re constructed by distributing F-POSS ( D 1nm ) rndomly on surfces t given prticle coverge rtio. On the other hnd, PP chins nd F-POSS prticles re plced in lrge periodic box t given prticle volume frction to construct the finl morphous PP nnocomposite. The coverge rtio is then clculted by the sttistic verges of prticles on the surfces with severl rndom cuts. 4. Determintion of the energy prmeter of PP The prerequisite of resonble force field is tht it cn well reproduce the chemicl potentil equilibrium inside nd outside of pure PP mtrix. The TrPPE-UA bonded nd nonbonded prmeters pper to well describe the structure of PP chins. When these prmeters re pplied to clculte phse coexistence curves, however, the results devite systemticlly from the experimentl dt. Thus it is necessry to regulte some prmeters depending on those experimentl mesurements. For simplicity, the intrmoleculr bonding prmeters re supposed to remin invrint, wheres the nonbonded energy prmeters re regressed using
the experimentl dt. 13 Fig. S1 shows the chemicl potentil vrition with incresing density vi the grnd cnonicl trnsformtion 14 using the new energy prmeter. After bout the tenth trnsform, the curve is visully flt in the rnge of trnsition, nd the two singulr points t both ends represent equilibrium densities outside nd inside the PP mtrix. Such prmeter is then pplied to the CO 2 /PP/F-POSS ternry systems without further regultion. Fig. S1. Trnsforming process for serching the chemicl potentil equilibrium outside nd inside PP mtrix. 5. Solubility CO 2 solubility plys n importnt role in PP foming, nd should be ddressed in dvnce. It is generlly ccepted tht crystllites re impenetrble for most non-rective molecules including CO 2. Smll molecules cn diffuse into morphous regions. The crystl domin is usully considered s inert filler becuse gs cnnot ccess. The CO 2 solubility properties in this work re referring to those in morphous PP mtrix. Fig. S2 presents the experimentl nd clculted pressure composition curves for CO 2 /PP t different tempertures. The verge devition between clculted nd experimentl vlues 15 is 3.70%. At the
beginning of dsorption, the CO 2 composition in PP phse exhibits liner correltion to the pressure. As the pressure increses, CO 2 molecules re forced to penetrte into PP mtrix. The influence of temperture on the composition could be explined by its effect on bulk CO 2 density. Fig. S2. Pressure composition curves of CO 2 (1) nd PP (2) mixture t different tempertures. x 1 is the mss frction of CO 2 in PP phse, nd y 1 is the mss frction of CO 2 in CO 2 phse. The spots re experimentl dt. 15 Fig. S3 illustrtes the density vritions, where the pressure declines from different initil pressures to the foming pressure, such s the mbient pressure (0.1MP). In the process of decompression, CO 2 mss frction is constnt, but the densities of CO 2 nd PP decrese. Nevertheless, the concentrtion of CO 2 is severl orders lrger thn the equilibrium concentrtion t the foming pressure. Therefore, the PP mtrix swells, resulting in further decrese of density nd n increse of swelling rtio. In this cse, the overswelled PP mtrix provides enough free volume for bubble nucletion nd growth. As consequence, CO 2 concentrtion shows supersturtion, but the PP concentrtion declines to substurtion, which re impertive for the metstble bubble nucletion.
Fig. S3. Pressure density curves in the systems of CO 2 /PP with different mss frctions of CO 2 in PP phse. The dotted lines switch to the foming pressure ( 0.1MP ). 6. Excess free energies We hve clculted the excess free energies for different PP/CO 2 binry systems in contct with the morphous or crystlline PP surfces. The results re shown in the following Tble S1. It is shown tht the contribution of conformtionl entropy ( F chin F conf ) exceeds tht of interction enthlpy hs tt ( F F ), showing the inherent chrcteristic of polymer systems. In ddition, the interction of crystl surfce exceeds tht of morphous surfce t given prticle coverge, resulting in reltively lower surfce tension. Tble S1 Excess free energies for the PP/CO 2 binry systems in contct with different composite surfces.
System Amorphous Crystlline Prticle coverge rtio% id F /k B T hs F F /k B T tt F F /k B T chin conf ext F /k B T 0.00 0.42 0.16-0.25-0.092 42.70 0.039 0.017-0.023-70.00 0.090 0.036-0.055-0.013 0.00 0.39 0.15-0.22-0.12 42.70 0.032 0.014-0.018-70.00 0.090 0.036-0.055-0.013
Notes nd references 1 Y. Rosenfeld, Phys. Rev. Lett., 1989, 63, 980. 2 Y.-X. Yu nd J. Wu, J. Chem. Phys., 2002, 117, 10156. 3 D. Zhou, J. Mi nd C. Zhong, J. Phys. Chem. C, 2012, 116, 3042. 4 S. Jin, A. Dominik nd W. G. Chpmn, J. Chem. Phys., 2007, 127, 244904. 5 K. S. Schweizer nd J. G. Curro, Phys. Rev. Lett., 1987, 58, 246. 6 K. G. Honnell, J. G. Curro nd K. S. Schweizer, Mcromolecules, 1990, 23, 3496. 7 M. Xu, J. Chen, C. Zhng, Z. Du nd J. Mi, Phys. Chem. Chem. Phys., 2011, 13, 21084. 8 A. Kovlenko nd F. Hirt, J. Chem. Phys., 1999, 110, 10095. 9 J. B. Hooper nd K. S. Schweizer, Mcromolecules, 2005, 38, 8858. 10 J. G. Curro nd A. L. Frischknecht, Polymer, 2005, 46, 6500. 11 G. J. Mrtyn, M. E. Tuckermn, D. J. Tobis nd M. L. Klein, Mol. Phys., 1996, 87, 1117. 12 D. Argyris, N. R. Tumml, A. Striolo nd D. R. Cole, J. Phys. Chem. C, 2008, 112, 13587. 13 J. Li, W. Cheung nd D. Ji, Polymer, 1999, 40, 1219. 14 Y. Tng, J. Chem. Phys., 2011, 134, 224508. 15 D.-c. Li, T. Liu, L. Zho nd W.-k. Yun, Ind. Eng. Chem. Res., 2009, 48, 7117.