ARTICLE
https://doi.org/10.1038/s41467-019-13410-z OPEN An unexpected N-dependence in the viscosity reduction in all-polymer nanocomposite
Tao Chen1,3, Huan-Yu Zhao 1,3, Rui Shi 1,3, Wen-Feng Lin1, Xiang-Meng Jia1, Hu-Jun Qian1*, Zhong-Yuan Lu1, Xing-Xing Zhang2, Yan-Kai Li2 & Zhao-Yan Sun2
Adding small nanoparticles (NPs) into polymer melt can lead to a non-Einstein-like decrease in viscosity. However, the underlying mechanism remains a long-standing unsolved puzzle.
1234567890():,; Here, for an all-polymer nanocomposite formed by linear polystyrene (PS) chains and PS single-chain nanoparticles (SCNPs), we perform large-scale molecular dynamics simulations and experimental rheology measurements. We show that with a fixed (small) loading of the SCNP, viscosity reduction (VR) effect can be largely amplified with an increase in matrix chain length N, and that the system with longer polymer chains will have a larger VR. We demonstrate that such N-dependent VR can be attributed to the friction reduction experienced by polymer segment blobs which have similar size and interact directly with these SCNPs. A theoretical model is proposed based on the tube model. We demonstrate that it can well describe the friction reduction experienced by melt polymers and the VR effect in these composite systems.
1 State Key Laboratory of Supramolecular Structure and Materials, Institute of Theoretical Chemistry, Jilin University, Changchun 130023, China. 2 State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China. 3These authors contributed equally: Tao Chen, Huan-Yu Zhao, Rui Shi. *email: [email protected]
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dding nanoparticles (NPs) into polymer melts can lead to found direct evidences showing that these SCNPs were dynami- fascinating improvements in materials mechanical1–4, cally coupled with polymer chain segment blobs with a similar A 5–7 8 optical , and electrical properties. It is both funda- size as these SCNPs. In order to understand the mechanism of mentally and practically important9,10 to understand the under- the substantial, nonlinear VR in all-PS composite system11,12, lying mechanisms of NPs manipulation on material properties. large-scale coarse-grained molecular dynamics simulations are However, due to complex influencing factors in the system, many performed in this study. The polymer/NP composite (PNC) experimentally reported peculiar phenomena are yet far from systems with different chain lengths and different volume frac- being fully explored. For instance, early seminal works by Mackay tions of SCNPs are systematically investigated. Interestingly, we and coworkers reported a dramatic reduction in viscosity of find an unexpected N-dependence for the acceleration in the PS melt due to the addition of a small amount of cross-linked chain relaxation dynamics. We also carry out experimental single-chain nanoparticles (SCNPs)11–13. This phenomenon is in rheology measurements of the corresponding all-PS composite a striking contrast to Einstein-Batchelor14,15 law which predicts a systems, predictions from simulations are confirmed, i.e., VR viscosity increase by incorporating NPs in the system. It also effect is found to be larger in systems with larger molecular remains so far a major unresolved puzzle in polymer field. weight for linear polymer chains. Based on both simulation and Earlier works can be traced back to the work of Malinskii and experimental results, we propose a theoretical model based on the coworkers in the 1970s, where an anomalous reduction of the tube model to understand the VR effect and its N-dependence. PVC melt viscosity caused by the addition of NPs was reported16, it was attributed to the creation of additional free volume at the Results polymer/NP boundary surface region16,17. However, it reverted to Influence of NPs on melt polymer chain conformation. First of a viscosity increase with a further increase in NP loading (ϕ). all, we calculate the average values of end-to-end distance Ree and Such a transition from viscosity reduction (VR) to increase radius of gyration R of polymer chains in both pure melt and fi g was con rmed recently in a poly(ethylene-alt-propylene)/silica composites with different NP loadings, the results are listed in nanocomposites18. A recent molecular dynamics simulation 19 Supplementary Table 2. It shows that the presence of NPs has study showed that with a small loading of NPs, VR effect can be almost no effect on the chain dimensions, except a very small attributed to the disentanglement effect found for melt polymer decrease in R and R at very hight NP loadings. The same results chains. However, NPs were found to be “entangled” when its ee g had been reported in other simulations, a nice review/discussion loading reached a critical value and hence melt chain relaxations on this topic can be found in a recent review article by Kröger and were very much hindered. Other than these reports, VR was cowokers29. The primitive path analysis19,30–32 using Z1 code30–32 also widely reported in the composite systems of polymers shows that the length of the primitive path L in PNC system with inorganic NPs13,20,21, organic fullerenes13,22, grafted silica pp NPs23,24, and dendrimers25. Various polymer and NP types in even at hight SCNP loading still follows Gaussian distribution these systems would result in different types of interactions. In (see Supplementary Fig. 2), which is in good agreement with 33 19 contrast, these complex enthalpic interactions do not exist in the experiment and other simulation results. The average value of hL i and its standard deviation σ are listed in Supplementary all-polystyrene (PS) composites reported by Mackay and cow- pp Lpp 11–13 h i orkers . Therefore all-PS composite can serve as an ideal Table 3. Note that although Lpp is reduced at high NP loadings, model system. In addition, the interface region in such all-PS the σ remains almost unchanged, which implies that NP has no Lpp composite system has a liquid-liquid contact between PS SCNPs effect on the fluctuation of L . Namely, it has nearly no effect on and melt polymers, which is also different from a solid-liquid like pp the contour length fluctuation (CLF) of the melt chains, which is interface existing in inorganic NP/polymer composites. From this believed to be crucial to the melt chain dynamics. point of view, a recently reported dendritic polyethylene (dPE)/PS nanocomposite25 has the same characteristic of soft interface. More importantly, adding a very small amount of either PS Entanglement between polymer chains in simulation. For the SCNPs or dPE into PS melts can cause a dramatic viscosity pure PS720 system, the average entanglement strand length fi 34 reduction: Tuteja et al.12 reported that the PS melt viscosity can obtained from modi ed coil estimator in primitive path analysis h i¼ : be reduced up to 80% by adding 1% PS SCNPs, while similarly is Ne 163 7, which is in a good agreement with experimen- addition of 5% dPE NPs25 in PS melt can cause a reduction of tally measured entangled molecular weight of 17 kDa, as listed in 26 melt viscosity up to 95%. Such an abrupt decrease in viscosity is the text book by Rubinstein and Colby (see in Table 9.1 on page obviously beyond conventional disentanglement effect, which is 362). Therefore PS720 chain has 4 entanglement strands per predicted to be proportional to the volume fraction of the NPs19. chain on average. In addition, we calculate the mean square ð Þ On the other hand, available experiment results12,25 showed that displacement (MSD) of the two innermost monomers (g1 t ) and 0 that of the center of mass (CM) (g ðtÞ) for PS720 chains. the plateau modulus (GN ) is nearly unaffected at low NP loadings, 3 indicating no obvious change in the melt chain entanglements The MSD is defined as gðtÞ¼hjrðtÞÀrð0Þj2i, where rðtÞ is the 0 ρ 26 ρ ð Þ according to GN ekBT , where e, kB, and T are the number coordinate of either the two innermost monomers for g1 t or the density of entanglements, Boltzmann constant and temperature, ð Þ CM of the PS chain for g3 t at time t. The results are shown in respectively. Therefore, the large reduction in viscosity in these Fig. 1. Although we do not plot the results at small time scale, systems cannot be simply explained by disentanglement effect. At g1(t) and g3(t) show different scaling relations in distinct the same time, according to Goldansaz et al.25, such VR cannot be 26,35 1=2 regions . For instance, g1(t) scales as t at short time scale attributed to confinement effect, free volume effect, surface slip- = t<τ , followed by a lower scaling of g ð t Þt1 4 in the tube page, shear banding, or particle induced shear thinning. e 1 τ τ ð Þ 1=2 τ ð Þ Our previous simulations27 with a single PS SCNP presented in ( e
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g (t) for PS250 1 1 > p = 1 a 2 g (t) for PS720 1 p 100 1 0.8 p = 2 )
2 g (t) for PS720 p = 3 3 p = 4
1/4 (0)>/ ( 0.4 t 3 10 ( g 1/2 p 1 200 1 ) 20% t ( g 1 150 g > b 100 2 p 50 0.8 0 0.1 0 1 × 107 2 × 107 Time (ps) (0)>/ 10 10 10 10 10 10 t ( Time (ps) p 2 systems with different NP loadings are small due to log scale we ) 2 10 adopted. Plots of the same data with linear scale are supplied in 101 = 0% Supplementary Fig. 3, which shows clearer differences. Short dash )> (nm t = 10% ( lines in Fig. 2 are MSD curves for NPs in representative PNC 2 r 0 = 20% < 10 systems with a SCNP loading of 20%, which shows NPs diffuse NP much faster than melt chains. It is interesting to see that SCNPs –1 10 seem to follow similar Rouse-like time dependent behavior as 103 b those of the CMs of the PS chains, i.e., there is a transition from a sub-diffusive behavior at short time scale to a Fickian regime at ) 2 2 10 long time scale, though this transition happens a little earlier for SCNPs. We attribute this similarity to the length-scale dependent 1 = 0% )> (nm 10 t ( = 10% interaction between these SCNPs and the matrix polymer chains. 2 r 28,37 = 20% fi < Speci cally, in our previous simulations , SCNPs are found to 100 NP interact directly with chain segment blobs which have similar size –1 as these SCNPs, and we observe that such chain segment blobs 10 4 5 6 7 have 144 monomers each and a little smaller size than entan- 10 10 10 10 glement strand. Therefore the transition from sub-diffusive to Time (ps) Fickian occurs a little earlier for SCNPs. h ð ÞÁ Fig. 2 Solid lines are the MSD of linear matrix PS chains in the composites In Fig. 3 we plot the Rouse mode analyses results of Xp t ϕ ð Þi=h 2i of a PS500/NP250 and b PS1300/NP250 with different loadings ( )of Xp 0 Xp for both PS500 and PS720 pure melts and composite NP250. The MSD for the CM of NPs are plotted as references (dash line). systems containing 20% NP250, here Xp is the Rouse mode as defined in Supplementary Eq. (1). Results with mode indices of system only has 4 entanglements per chain. For the monomer ¼ à p 1 4 are plotted, which are the slowest relaxation modes in MSD curve, a transition is found at τ 20 ns from short time our simulated systems. It shows that these modes have been fully ð Þ 1=2 β scale behavior with g1 t t to intermediate time scale with relaxed. We plot in Supplementary Fig. 3 the values of p ð Þ 1=4 τ ¼ : g1 t t . Therefore e 23 2 ns according to the relation (see Supplementary Eq. (4)) as a function of N=p for the first 20 τ ¼ 36 τà 36 β e π3 given by Likhtman and McLeish . In addition, g1(t) is modes for pure PS melts with different N. The minimum of p is = also calculated for PS250 and shown in Fig. 1. Although we know found at 166, which is close to N p Ne and is consistent with that entanglement in this system is very weak since each chain has the results as reported in ref. 19 in the aspect that the minimum of = fi N Ne < 2 entanglements, a de nite decrease of the slope in the β appears at the entanglement length. More importantly, results τ τ p intermediate regime is observed after e. Before e, it has exactly in Fig. 3 show that the relaxation of chains in the composite the same behavior as in PS720 system. system (dashed color lines) are much faster than those in pure melts (black lines). Acceleration in polymer chain dynamics. To characterize the translational dynamics of polymer chains, we calculate the MSD Unexpected N-dependence in the acceleration of chain of the CM of polymer chains in both composite and pure polymer dynamics. From the Rouse mode analysis, we obtain the τ eff melt systems. The results are shown in Fig. 2. The linear PS relaxation times p at different p modes, which correspond to chains in two systems are 500 or 720 styrene monomers long and the effective relaxation time of melt chain segments with N=p indicated as PS500 and PS720 respectively. In both systems, monomers, details can be found in Supplementary Eq. (4). The MSDs of PS chains in composite are faster than those in pure results are plotted in Fig. 4a for systems with different chain melt. Note that the differences between results from composite lengths but a fixed NP loading of 20% and the results are NATURE COMMUNICATIONS | (2019) 10:5552 | https://doi.org/10.1038/s41467-019-13410-z | www.nature.com/naturecommunications 3 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-13410-z 1.0 1.0 a PNC/pure,Simulation 0.9 0.8 = 0) ( PS250 PNC pure p eff 0.6 / / 0.8 PS350 experiment p eff PS500 Ratios 0.4 0.7 PS720 100 0.2 PNC pure N/p / ,ref.12 0.0 100 1000 1.0 b Molecular weight (kDa) Fig. 5 The relative viscosity ηPNC=ηpure for the systems with different PS = 0) 0.9 PNC pure molecular weight at a fixed NP loading of 2% at T = 170 C, η and η ( eff are the viscosities of PNC and pure PS melt systems respectively. NP has a / 0.8 eff PNC2% molecular weight of 26.2 kDa. The last data point in purple is taken from PNC10% ref. 12, where NP loading is 1% and SCNP has a molecular weight of 25 kDa. 0.7 PNC20% Triangles are the ratio between effective terminal relaxation times for 200 300 400 500 600 700 800 composite systems with a SCNP loading of 2% and that of the pure melt, N this ratio is taken from Fig. 4aatp = 1. Fig. 4 Effective relaxation time ratio between composite and neat supplied in the section of Experimental details in Supplementary system. a The ðN=pÞ-dependent segmental effective relaxation time τ eff p Information. obtained from Rouse mode analyses is plotted for matrix chain segments The first result we look at is the scaling of zero-shear visocity = with an effective length of N p in composite systems containing 20% with respect to N in pure polymer melt without any NPs, the = NP250, only values with N p > 50 are plotted; b the N-dependent chain results are plotted in Supplementary Fig. 4, which shows a scaling τeff ð Þ ð Þ ð Þ effective relaxation time obtained from Rheology measurements. In order to testify the predictions from Disentanglement effect. The number of entanglements per chain the simulation, experimental rheology (small amplitude oscilla- h i tory shear) measurements are performed. The composite systems Zkink obtained from Z1 analysis is plotted in Fig. 6 versus τeff are composed of linear polystyrene chains with molecular weights relaxation time ( ) of the melt chains at various NP loadings, of 54.6, 98.5, and 201.8 kDa, and PS SCNPs with a molecular the corresponding data are listed in Supplementary Table 4. Both h i τ weight of Mw = 26.2 kDa. Linear polystyrenes were purchased values of Zkink and eff are normalized by the corresponding ϕ ¼ from Polymer Source Inc., they were purified via a reprecipitation values obtained in pure melt systems at 0. Note that Rouse 19,40–42 – process before use. PS SCNPs were synthesized following the mode analysis (see Supplementary Eqs. (1 4)) is per- eff procedure proposed by Hawker et al.38. Details of the synthesis formed to calculate the relaxation time τ ðpÞ at different and characterization procedures for PS SCNPs, preparation of p modes, from which τeff ðp ¼ 1Þ is taken as the chain relaxation the composites, system details and rheology measurements are time τeff . Upon increase of NP loading (ϕ) in the system, we do 4 NATURE COMMUNICATIONS | (2019) 10:5552 | https://doi.org/10.1038/s41467-019-13410-z | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-13410-z ARTICLE 6 0% 10 0% 1.0 5% (0%) 5 d 20% 10% 10 10% = 0) 2% 30% ( 0.8 40% 4 d(20%) eff 10 )/ 50% 20% ( Slop = 1.03 eff 30% 3 0.6 10 Slop = 1.82 G' in pure PS 40% 102 G'' in pure PS 0.4 G' in PNC Storage and loss modulus (Pa) Storage 0.6 0.7 0.8 0.9 1.0 G'' in PNC 1 fi h i ¼ : nd a disentanglement effect in the system, i.e., Zkink value with Ne 163 7 obtained from Z1 calculation in our simulation. decreases with the addition of NPs. At the same time, a decrease More importantly, we see that there is a large right shift of the τ 0 ¼ 00 is also found in the chain relxation time d. However, a stronger point for G G for the composite system with respect to that of τeff h i pure melt, which indicates a large reduction in disentanglement dependence of over Zkink is found in composite system with N = 720 (‘PS720/NP250’) than that with N = 500 (‘PS500/ time, which is consistent with the observed VR. However, the NP250’). Least squares fittings result in slopes of 1.03 and 1.82 for composite system has an almost identical plateau modulus (G0) systems PS500/NP250 and PS720/NP250 respectively, as shown as pure melt system. It indicates that the two systems have almost fi in Fig. 6. Intuitively, disentanglement effect caused by the pre- identical entanglement density in the system. This result con rms sence of NPs could be a direct possible reason for the acceleration that the mechanism of disentanglement effect is not the one in the chain dynamics. According to the tube model, chain responsible for the VR. τeff τ N τ τ relaxation time R RZ, where R is the Rouse time Ne Molecular mechanism. To the best of our knowledge, such of the chain and Z ¼ N is the chain entanglement density. Ne N-dependent acceleration in entangled polymer chain dynamics h i h 0 i = τeff ðϕÞ τeff ðϕ ¼ Þ Therefore, a simple relation Zkink / Z kink / 0 has not been reported before. The results in the above section can be derived since the addition of the NPs in the system has show that the disentanglement effect cannot be used to explain almost no influences on the chain dimension of the melt poly- the abrupt reduction in viscosity. To explore the underlying fl mers (see listed Ree and Rg values in Supplementary Table 2 for molecular mechanism, a full understanding of the in uence by melt polymer chains), a similar use and derivation of this relation SCNP on surrounding chain dynamics is crucial. Our previous 27 can be found in ref. 19. Although this relation is valid for com- simulations have demonstrated that such internally cross-linked posite system ‘PS500/NP250’ (squares in Fig. 6), a stronger SCNPs are soft in nature, and the dynamics of surrounding dependence is found in composite system ‘PS720/NP250’ (circles polymer chain segments are largely accelerated. Such acceleration in Fig. 6) where the reduction in the relaxation time is found effect at the SCNP surface area can be understood since the much faster than the reduction in entanglements. For instance, a SCNPs are identical to polymer chains in its composition but NP loading of ϕ = 20% can only cause a reduction of 10% in internally cross-linked, therefore the internal degrees of freedom h i % τeff are largely reduced for these SCNPs, which will result in a Zkink but a much larger reduction of 30 in is found. Note that a linear fitting in Fig. 6 for system ‘PS720/NP250’ is reduction in friction strength felt by surrounding polymer chains, only guided to the eye, we do not claim a rigorous use of this consequently a speed-up of surrounding chain dynamics can be relation. In addition, we have also simulated a PS720 nano- expected. In addition, interpenetration/contacts between sur- composite system which contains only 2% NPs. The relaxation rounding melt chain segments in the interior of the SCNP will be much reduced comparing with those between free melt polymer time ratio is found to be τeff /τeff ðϕ ¼ 0Þ = 88%, while the hZ i kink chains. To provide a direct evidence, we calculate the MSD of the remains unchanged. According to these results, we conclude that two inner most monomers on melt chain PS720 in both com- disentanglement effect cannot fully account for the acceleration in posite and pure melt systems, results are shown in the inset of chain dynamics found in the system. Here we have to note that Fig. 1. It clearly shows a speed-up effect in the composite system. according to Li and coworkers19, adding solid NPs can cause Such an acceleration effect directly indicate that frictions from confinement effects at large NP loadings. We do not observe such SCNPs on surrounding melt chain monomers are reduced com- confinement effects even at very high loadings of SCNPs, this can paring with frictions between free polymer chain monomers in be attributed to the inherent softness of these SCNPs. pure melt, note that the composite systems have almost the same Experimentally, we plot the measured results of storage and 0 00 density as pure melt. loss modulus (G and G ) for both pure melt and composite On the other hand, the SCNP has generally a faster diffusion systems in Fig. 7. Here the melt chain has a molecular weight of than the melt chain since the SCNP has an overall spherical shape 202 kDa and the composite has an NP loading of 2%. For pure on average and a smaller molecular weight. Representative MSD polymer melt system, the entanglement molecular weight is Me ρ curves of NP250 in PS500 and PS720 melts are plotted as dash 16.38 kDa according to G ¼ 4 RT, this value is very close to a 0 5Me lines in Fig. 2, SCNPs are found to diffuse much faster than the = 26 43,44 previously reported experimental value of Me 17 kDa .In melt chains. In addition, recent theoretical works predicted fi addition, the value of Me 16.38 kDa is in a good agreement that the NP will have a constant diffusion coef cient in long chain NATURE COMMUNICATIONS | (2019) 10:5552 | https://doi.org/10.1038/s41467-019-13410-z | www.nature.com/naturecommunications 5 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-13410-z melts where the NP is smaller than both radius of gyration and of Eq. (4) as, fi tube diameter of the melt chain, which is also veri ed in our XZ recent simulations37 for the current all-PS composite system. It ðϕ; Þ¼1 ðÀδϕAÞ : 1 ð Þ f Z exp Z ´ Z 5 can be attributed to the fact that when SCNP diffuses in polymer Z i¼1 melt, it only feels the friction from the surrounding polymer This function defines the dependence of friction coefficient on ϕ chain segments which have similar size with SCNP. Therefore, and Z. As a consequence, the curvilinear diffusion coefficient in when the melt chain length increases at a fixed loading of the Eq. (4) will be, SCNPs, the diffusion coefficient of SCNP does not change although that of the chain decreases cubically with N according to k T D ¼ B : ð6Þ the tube model. These results imply that interactions between c Zζ0f ðϕ; ZÞ SCNP and melt polymers are on the length scale of the SCNP 28 Eventually, the polymer chain relaxation time can be written size. Our previous simulations also provide direct evidences for 45 a dynamic coupling between these SCNPs and melt polymer as , chains on such a length scale, i.e., for polymer chain segments L2 τ ¼ ¼ 3τ Z3f ðϕ; ZÞ; ð7Þ having a similar size with SCNP. Note that these segments rep π2D e contain 144 styrene monomers each, which is smaller than but c τ ¼ similar with an entanglement strand, Ne, of melt polymers. where L is the length of the primitive path of the tube, e 2 2ζ0= π2 From the above analyses, we know that SCNP in the system Ne b 3 kBT is the Rouse time of the entanglement stands. interacts with melt PS chains on a length scale of SCNP size After considering the term for CLF effect, it turns out to be, qffiffiffiffiffiffiffi or entanglement strand. Knowing such a detailed interaction hiÀÁ 2 τ ¼ τ 3 À μ 1 ðÞ¼ϕ; τpure ðϕ; Þ: ð8Þ mechanism between SCNPs and free melt polymer chains will rep 3 eZ 1 Z f Z rep f Z be also the key to understand the observed abnormal NÀdependence in chain acceleration or friction reduction. The Similarly, the viscosity can be written as, sffiffiffiffiffiffiffiffiffiffi discussions in the following will be focused on the scale of "#3 π2k T 1 entangled strands, they are also referred to as blobs in the η ¼ B τ 3 À μ ðϕ; Þ¼ηpure ðϕ; Þ: ð9Þ ν eZ 1 f Z f Z following. 4 0Ne Z As we discussed above, interactions from SCNPs will effectively τpure ηpure reduce the frictions and therefore accelerate the dynamics of In the above Eqs. (8) and (9), rep and are the original fi surrounding blobs. For instance, if a blob i interacts with an de nition of chain relaxation time and viscosity for entangled SCNP the friction of the blob i will be reduced and therefore its chains of the tube model. dynamics will be accelerated. Such acceleration effect will not According to our analyses in the above section, a small amount disappear immediately while the SCNP leaves, instead it will also of SCNP loaded in the system has almost no effect on the chain effectively accelerate the dynamics of the neighboring blobs along conformations, i.e., composite system has an almost identical Ne the chain contour due to the chemical connectivity between these value as pure system. This is also verified in the analysis in Fig. 7. blobs. Namely chain blobs are dynamically correlated along the Therefore we have ηPNCðϕÞ=ηpure ¼ f ðϕ; ZÞ according to Eq. (9). chain contour. Based on the above consideration, we simply use a For the currently investigated composite systems, this ratio is summation in the following equation to take account of the plotted in Fig. 8. A least squares fitting of these data points using contribution of each j blob on the same chain to the variation of the relation in Eq. (9) results in parameters ϕδ ¼ 0:0792 and the friction on blob i at a given SCNP fraction ϕ, α ¼ 0:0759. It corresponds to a reasonable correlation length of 1=α 13 entanglement blobs. Note that in the form of Eq. (5), XZ d ζ ðϕÞ A is a matrix and i ¼ 1; 2; ÁÁÁ ; Z is the integers indicating the i ¼ ÀδA ζ ðϕÞ; ð Þ ϕ ij j 1 number of entanglements per chain. While for polymers in our d j¼1 experiment, we have decimal numbers of Z. Therefore fitting in ζðϕÞ is the friction constant experienced by blobs at a given SCNP Fig. 8 is already quite good, although experimental data points are loading ϕ, it converges to ζ0 in pure polymer melt with the not exactly located on the fitting curve. A schematic depiction of absence of SCNPs at ϕ ¼ 0. δ describes an effective reduction in such interactions between SCNP and polymer chain segment blob ζ ¼ = is drawn in Fig. 8 on the right. i, Z N Ne is the number of entanglements per chain, Aij With the parameters of ϕδ ¼ 0:0792 and α ¼ 0:0759, we plot describes the correlation between blobs or the contribution from fi ζ ð Þ=ζ neighboring j blobs along the chain backbone to blob i. Eq. (1) friction coef cient ratios, PNC ib 0, for our experimental can be rewritten as, systems with diferent molecular weights in Fig. 9 using the fi ζ ð Þ ζ fi de nition in Eq. (3). PNC in and 0 are friction coef cient on d ζ ¼ÀδAζ; ð2Þ blobs in the composite and pure PS melt respectively, ib is the d ϕ index of blobs along the chain contour. We see that there is a N-dependence in this ratio, system with larger molecular weight A :¼ ζ :¼ðζ ; ζ ; ÁÁÁ; ζ ÞT with Aij and 1 2 Z . It has a generic solution has a larger reduction in the friction, corresponding to a larger of, VR effect we found. Also it changes along the chain as we 0 expected, termial blobs have a lower reduction in friction than the ζ ¼ expðÀϕδAÞζ ð3Þ inner ones due to a dynamical decay of the acceleration effect Therefore, curvilinear diffusion coefficient of the polymer chain caused by the interaction between SCNP and these blobs. in the tube can be written as, Discussion D ðϕÞ¼PkBT ¼ P kBT : ð Þ c Z ζ ðϕÞ Z ðÀϕδAÞζ0 4 In this work, by performing large-scale molecular dynamics ¼ i ¼ exp i 1 i 1 simulations and experimental rheological measurements, Suppose correlation between blobs Aij decays exponentially dynamics in an all-polymer composite system composed of linear ¼ ðÀαj À jÞ =α on chain backbone, Aij exp j i , where 1 is the polystyrene chains and internally cross-linked SCNPs are inves- correlation length between blobs. If we define the denominator tigated. We demonstrate that adding soft SCNPs can dramatically 6 NATURE COMMUNICATIONS | (2019) 10:5552 | https://doi.org/10.1038/s41467-019-13410-z | www.nature.com/naturecommunications NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-019-13410-z ARTICLE 1.0 Expt. data Fitting with Eq.(9) 0.8 H c c H H n 0.6 pure / PNC 0.4 SCNP 0.2 Segment144 0.0 1 10 100 1000 z Fig. 8 Experimentally measured viscosity ratio between composites and pure PS melt, ηPNC/ηpure, plotted as a function of Z. The green line indicates a least squares fitting of these data points using Eq. (9). On the right is a schematic depiction of the length scale of the interactions between SCNP (in red) with polymer chain. The entanglement blob which has a similar size with SCNP is plotted in green. 1.0 described. We hope that our results cannot only shed light on the PS54 kDa understanding of VR in the polymer/NP composites and provide PS98 kDa new insights for the development of polymer theory for com- 0.8 PS202 kDa posite systems, but also can be helpful for the relevant material 0 ζ design. / ) b ib i ( 0.6 PNC Data availability ζ The data that support the findings of this study, including the Supplementary Information, are available from the corresponding author on request. 0.4 Received: 12 September 2018; Accepted: 1 November 2019; 0 2 46810 12 iibb fi ζ ð Þ=ζ Fig. 9 Friction coef cient ratios, PNC ib 0, plotted as a function of the fi index of entanglement blobs (ib) along the chain with the de nition in References Eq. (3) and the fitting parameters of ϕδ ¼ 0:0792 and α ¼ 0:0759 from 1. Gupta, S., Zhang, Q., Emrick, T., Balazs, A. C. & Russell, T. P. Entropy-driven ζ ð Þ ζ fi Fig. 8. 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Commun. 6, 7198 (2015). China for fruitful discussions. 24. Kim, D., Srivastava, S., Narayanan, S. & Archer, L. A. Polymer nanocomposites: polymer and particle dynamics. Soft Matter 8, 10813 (2012). Author contributions 25. Goldansaz, H. et al. Anomalous rheological behavior of dendritic H.J.Q. conceived the project and designed the both simulation and experiment. T.C. nanoparticle/linear polymer nanocomposites. Macromolecules 48, 3368 carried out the simulations, part of the rheology measurements and data analysis. Huan- (2015). Yu Zhao carried out experiments of synthesis and characterizations. R.S. derived the 26. Rubinstein, M. & Colby, H. Polymer Physics (Oxford University Press, Oxford, theoretical model and participated in the analysis of the simulation data. W.F.L., X.X.Z., 2003). Y.K.L., and Z.Y.S. carried out part of the rheology measurements. X.M.J. participated in 27. Chen, T., Qian, H.-J., Zhu, Y.-L. & Lu, Z.-Y. Structure and dynamics the analysis of the simulation data. T.C. and H.J.Q. co-wrote the manuscript with the properties at interphase region in the composite of polystyrene and cross- input from other co-authors. H.J.Q. and Z.Y.L. co-supervised the project. All authors linked polystyrene soft nanoparticle. Macromolecules 48, 2751 (2015). discussed the results and commented on the manuscript. 28. Chen, T., Qian, H.-J. & Lu, Z.-Y. Diffusion dynamics of nanoparticle and its coupling with polymers in polymer nanocomposites. Chem. Phys. Lett. 687, 96–100 (2017). Competing interests 29. Karatrantos, A., Clarke, N. & Kröger, M. Modeling of polymer structure and The authors declare no competing interests. conformations in polymer nanocomposites from atomistic to mesoscale: a review. Polym. Rev. 56, 385–428 (2016). Additional information 30. Karayiannis, N. Ch. & Kröger, M. Combined molecular algorithms for the Supplementary information is available for this paper at https://doi.org/10.1038/s41467- generation, equilibration and topological analysis of entangled polymers: 019-13410-z. methodology and performance. Int. J. Mol. Sci. 10, 5054–5089 (2009). 31. Kröger, M. Shortest multiple disconnected path for the analysis of Correspondence and requests for materials should be addressed to H.-J.Q. entanglements in two- and three-dimensional polymeric systems. Comput. Phys. Commun. 168, 209–232 (2005). Peer review information Nature Communications thanks Krishnamurthy Jayaraman 32. Shanbhag, S. & Kröger, M. Primitive path networks generated by annealing and the other, anonymous, reviewer(s) for their contribution to the peer review of this and geometrical methods: insights into differences. Macromolecules 40, work. Peer reviewer reports are available. 2897–2903 (2007). 33. Schneider, G. J., Nusser, K., Willner, L., Falus, P. & Richter, D. 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Macromolecules 35, 6332–6343 (2002). adaptation, distribution and reproduction in any medium or format, as long as you give 37. Chen, T., Qian, H.-J. & Lu, Z.-Y. Note: Chain length dependent nanoparticle appropriate credit to the original author(s) and the source, provide a link to the Creative diffusion in polymer melt: effect of nanoparticle softness. J. Chem. Phys. 145, Commons license, and indicate if changes were made. The images or other third party 106101 (2016). material in this article are included in the article’s Creative Commons license, unless 38. Harth, E. et al. A facile approach to architecturally defined nanoparticles via indicated otherwise in a credit line to the material. If material is not included in the intramolecular chain collapse. J. Am. Chem. Soc. 124, 8653 (2002). article’s Creative Commons license and your intended use is not permitted by statutory 39. Mackay, M. E. & Henson, D. J. The effect of molecular mass and temperature regulation or exceeds the permitted use, you will need to obtain permission directly from on the slip of polystyrene melts at low stress levels. J. Rheol. 42, 1505 the copyright holder. To view a copy of this license, visit http://creativecommons.org/ (1998). licenses/by/4.0/. 40. Kalathi, J. T., Kumar, S. K., Rubinstein, M. & Grest, G. S. Rouse mode analysis of chain relaxation in homopolymer melts. Macromolecules 47, 6925 (2014a). © The Author(s) 2019 8 NATURE COMMUNICATIONS | (2019) 10:5552 | https://doi.org/10.1038/s41467-019-13410-z | www.nature.com/naturecommunications