Viscosity Overshoot in Biaxial Elongational Flow: Coarse-Grained Molecular Dynamics Simulation of Ring-Linear Polymer Mixtures
VViscosity Overshoot in Biaxial Elongational Flow:Coarse-Grained Molecular Dynamics Simulationof Ring-Linear Polymer Mixtures.
T. Murashima, ∗ , † K. Hagita, ‡ and T. Kawakatsu † † Department of Physics, Tohoku University, 6-3, Aramaki-aza-Aoba, Aoba-ku, Sendai,980-8578, Japan ‡ Department of Applied Physics, National Defence Academy, 1-10-20, Hashirimizu,Yokosuka, 239-8686, Japan
E-mail: [email protected]
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Abstract
Viscosity overshoot of entangled polymer melts has been observed under shearflow and uniaxial elongational flow, but not under biaxial elongational flow for a longperiod of time. We confirmed the presence of the viscosity overshoot under biaxialelongational flows observed in a mixed system of ring and linear polymers expressedby coarse-grained molecular dynamics simulations. The overshoot was found to bemore pronounced in a weakly entangled melt. Furthermore, the smallest strain rateshowing the strain hardening was found to be depending on the linear chain length as˙ ε s ( N ) ∼ N − / , which is different from the conventional relationship, ˙ ε s ( N ) ∼ N − or N − . for melts of linear polymers without or with entanglements, respectively.We have concluded that the cooperative phenomena of ring and linear chains wereenhanced under biaxial elongational flow. a r X i v : . [ c ond - m a t . s o f t ] F e b ntroduction Entangled polymer melts exhibit nonlinear flow behavior depending on their molecular struc-tures and compositions.
It is well known that many entangled polymers show viscosityovershoot under shear flow.
The branched polymers show a viscosity overshoot under uni-axial elongational flow. However, no viscosity overshoot behavior has been reported for along time under biaxial (or equibiaxial) elongational flow.There are various debates regarding the origin of viscosity overshoot.
Under shear flow,entanglements among polymers result in a temporary excess orientation of the subchain ofpolymer. Viscosity is attenuated by the relaxation of the overly oriented subchains. Theviscosity becomes a steady state when the forced orientation by the outside field and therelaxation of excess orientation are balanced.
Another opinion is that this is not due toorientation, but to the temporary elongation of polymers.
The viscosity overshoot of thebranched polymers seen under uniaxial elongational deformation is understood by the POM-POM model. Under uniaxial elongational deformation, the backbone is first stretched inthe elongational direction and the tension is increased. Then, as the arms of the branchbegin to be drawn into the backbone tube, the tension decays. Once the branched arms arecompletely inside the backbone tube, the viscosity shows a steady state, and the branchedpolymer behaves similar to a linear polymer. To make it happen the viscosity overshootunder biaxial elongational flow, there should be a mechanism depending on the moleculararchitecture to maximize a tension in a two-dimensional plane parallel to the elongationaldirections.Ring polymers are candidate to exhibit viscosity overshoot under biaxial elongationaldeformation due to the maximization of tension in the two-dimensional plane. Recently,ring polymers are getting a lot of attention because it has been found that the mechanicalresponse of ring polymers is quite different from that observed in conventional polymersystems. The stress relaxation of ring polymers exhibits a power law decay on the long-timeside due to a phenomenon called ”threading”, where the polymers penetrate each other. It has alsobeen reported that when a small amount of ring polymer is added to the linear polymer melt,the relaxation time increases, and the viscosity increases in proportion to the concentration ofthe ring polymer.
Under shear flows, hydrodynamic interaction has inflated ring polymersin a solvent and a linear chain matrix. The inflation of the DNA-ring polymer has beenobserved in semidilute linear polymer solutions under elongational flow. In molecular dynamics simulations, it has been difficult to handle large deformation ofelongation, except for planar elongation, due to the problem of boundary conditions. TheKraynik-Reinelt (KR) boundary condition, which has been used for planar elongationaldeformation, has been extended by Dobson and Hunt to handle uniaxial elongationaldeformation and equibiaxial elongational deformation in molecular dynamics simulations.The simulation code was provided by Nicolson and Rutledge, which led to vigorous studiesof extensional flow of polymer melts. Especially for ring polymer melts, O’Conner etal. found anomalous thickening behaviors owing to “reef knots” or persistent knots madeof two rings in uniaxial elongational flow. Very recently, Borger et al. found threading-unthreading transition for ring-linear blends in uniaxial elongational flow, where persistentknots were suppressed to form in ring-linear blends. Since the study of ring-linear blendsin elongational flows is just beginning, it is expected to find new phenomena in the otherelongational flows.In this study, we focus on ring-linear blends in biaxial elongational flow. The ring fractionof blends studied here is set to less than 0.1. This fraction is smaller than that in theprevious work where the uniaxial elongational flow has represented the stress overshootphenomena. The melt mixtures of ring and linear polymers under biaxial elongationalflow were analyzed by coarse-grained molecular dynamics simulations, and the changes instress or viscosity over time were investigated. We have visualized the rings and obtainedthe number of penetrations under biaxial elongational flows. From these analyses, we havefound the cooperative phenomena of ring and linear chains under biaxial elongational flows.3 imulation Methods
Kremer-Grest type bead-spring model
Ring and linear polymers were represented by a bead-spring model, namely the Kremer-Grestmodel. Ring and linear polymers are composed of N R and N L particles connected with N R and N L − M R and M L , respectively. The polymers are placed in a cubic unit cell L = diag( L, L, L )with periodic boundary conditions, where L = V / and V is the volume of unit cell.The dynamics of particles in equilibrium is governed by Newton’s equation of motion. d r dt = v , (1) m d v dt = F , (2)where r is the position, v is the velocity, and m is the mass of particle. The force F iscomposed of a conservative force, a viscous force, and a random force, F = − ∇ U − Γ v + (cid:112) k B T R ( t ) , (3)where Γ is the dumping coefficient, k B is Boltzmann’s constant, T is the temperature. Therandom vector R ( t ) has a Gaussian probability distribution function with correlation func-tion (cid:104) R i ( t ) R j ( t (cid:48) ) (cid:105) = δ ij δ ( t − t (cid:48) ). Potential energy U ( r ) consists of a Lennard-Jones (LJ)potential U LJ ( r (cid:48) ) among intramolecular and intermolecular particles and a finite extensible4onlinear elastic (FENE) potential U FENE ( r (cid:48)(cid:48) ) for particles connected with a bond; U ( r ) = (cid:88) intra + inter U LJ ( r (cid:48) ) + (cid:88) intra U FENE ( r (cid:48)(cid:48) ) (4) U LJ ( r ) = (cid:15) (cid:20)(cid:16) σr (cid:17) − (cid:16) σr (cid:17) (cid:21) + (cid:15), ( r < r c )0 , ( r ≥ r c ) (5) U FENE ( r ) = − k R ln (cid:20) − (cid:18) rR (cid:19)(cid:21) , ( r < R )0 , ( r ≥ R ) (6)where (cid:15) is the unit of energy, σ is the size of particle, r c is a cutoff length of LJ potential, k is the bond coefficient, and R is the maximum extent of the FENE bond. To investigatethe rheological properties, we observe the stress tensor σ αβ = − V N total (cid:88) i mv αi v βi + N (cid:48) total (cid:88) i r αi F βi , (7)where N total is the total number of LJ particles in the system, N (cid:48) total includes the periodicimage particles.We set the parameters to the conventional ones; Γ = 0 . m/τ , T = 1 . (cid:15)/k B , r c = 2 / σ , k = 30 . (cid:15)/σ , R = 1 . σ , number density ρ = N total /V = 0 . /σ , unit of time in LJ system τ = (cid:112) mσ /(cid:15) , and LJ unit σ = m = (cid:15) = k B = 1.5 eneralized Kraynik-Reinelt boundary condisions under elonga-tional flows In a flow field K = ( ∇ v ) T , Newton’s equation of motion is modified to the SLLOD equa-tion. d r dt = v + K · r , (8) d v dt = F m − K · v . (9)At the same time, the unit cell L = ( e , e , e ), which is not necessarily equal to diag( L, L, L ),is also deformed owing to the flow field K . d L dt = K · L . (10)Then, we find L ( t ) = exp ( K t ) L (0) = ( e ( t ) , e ( t ) , e ( t )) . (11)In general, the shape of L ( t ) is a parallelepiped. If a pair of parallel faces becomes very closeand the gap between the faces is less than r c , the simulation will collapse. If we can find aunit cell ˜ L = ( ˜ e , ˜ e , ˜ e ), which is close to a cubic shape or a rectangular parallelepiped, andthe unit cell L ( t ) satisfies L ( t ) = ( n ˜ e , n ˜ e , n ˜ e ) , ( n i ∈ Z ) (12)then we can reduce L ( t ) to ˜ L . The LLL-algorithm and Semaev’s algorithm are useful tofind ˜ L .The flow field K of biaxial elongational flow with elongational rate ˙ ε is represented as6 = diag( ˙ ε, ˙ ε, − ε ). Then, we find L ( t ) = diag (cid:0) e ˙ εt , e ˙ εt , e − εt (cid:1) L (0) . (13)If we choose L (0) = diag( L, L, L ) as usual, the unit cell L ( t ) = diag (cid:0) Le ˙ εt , Le ˙ εt , Le − εt (cid:1) isflattened for finite time steps without being able to find ˜ L . To avoid this, we choose aninitial unit cell as L (0) = V − = .
737 0 .
591 0 . − .
328 0 . − . − .
591 0 .
328 0 . . (14)The column vectors of V − are the eigenvectors of automorphism M ∈ SL(3; Z ) such that L ( t + t ) = M · L ( t ). Here, SL(3; Z ) means the special linear group over integers of degreethree, which is the group of volume and orientation preserving linear transformations of Z .In this case, we can find a reduced unit cell ˜ L , and then we continue to apply the elongationalflow to the system for infinitely long time steps.In practice, it is convenient to choose an upper triangular unit cell L MD in MD simulationutilizing spatial-decomposition techniques for parallel computing. We can convert L =( a , b , c ) to L MD as L MD = a · ˆ e b · ˆ e c · ˆ e b · ˆ e c · ˆ e c · ˆ e , (15)where ˆ e = ˆ a = a / | a | , ˆ e = (ˆ a × ˆ b ) × ˆ a , ˆ e = ˆ a × ˆ b . In other words, L and L MD arerelated through a rotation matrix Q ; L = Q · L MD . The rotation matrix Q is obtained by Q = L · L − . The flow field K is applied on L -frame, and then the positions, velocities andforces are updated on L MD -frame in simulation algorithm. imulation setups Here, we consider ring with 160 beads (called as R160) and linear chains with 10, 40, and160 (L10, L40, and L160). We investigate 1:10 blends of ring and linear polymers (R160-L10, R160-L40, and R160-L160). The total number of beads in the system M R N R + M L N L is fixed to 675,840, and then M R160 = 384, M L10 = 61 , M L40 = 15 , M L160 =3 , . × equilibration runs or 1 . × τ , which is much larger than the longest relaxation time ofL160, τ , L160 (cid:39) . × τ (see Ref. ). We have confirmed in our recent work that 1 . × steps at least are required for achieving equilibrium distribution of linear chain penetrationthrough rings. For long-time equilibration, we used LAMMPS and HOOMD-blue. To apply a biaxial elongational flow with the elongational flow rate ˙ ε to the equilibratedsystem, we used LAMMPS with USER-UEFEX. The biaxial elongational flow rates wereselected as ˙ ε = { × − , × − , × − , × − , × − , × − , × − } τ − , whichincludes the inverse of the entanglement time τ e (cid:39) ) and the inverse of themaximum relaxation time τ , L160 in this study, and nonlinear behaviors can be captured.Each elongational flow simulation was conducted on 1 . × steps. Results and Discussion
Relaxation modulus (linear viscoelasticity)
Before going to discuss the biaxial elongational flows, we would like to summarize the staticproperties of ring-linear blends. From the Green-Kubo relation on stress auto-correlationfunctions in equilibrium states, we can obtain the relaxation modulus, G ( t ) = Vk B T (cid:104) σ xy ( t ) σ xy (0) (cid:105) , (16)8igure 1: Relaxation modulus of pure linear melts (L10, L40, L160), pure ring melt (R160),and ring-linear blends (R160-L10, R160-L40, R160-L160). The one-eleventh values of R160were plotted for guide to eyes.where the stress auto-correlation functions are calculated by the multiple-tau method. Figure 1 summarizes the relaxation moduli of L10, L40, L160, R160, R160-L10, R160-L40,and R160-L160. The one-eleventh of the relaxation modulus of R160 is also plotted in Fig.1 for guide to eyes.The relaxation modulus of R160-L10 clearly shows two step relaxation, while those ofR160-L40 and R160-L160 do not show. The relaxation moduli of R160-L40 and R160-L160are slightly deviating from those of L40 and L160, respectively. However, the deviation isnegligibly small. We found that the ring polymers, R160, introduces slow modes into linearmatrix of L10 whereas the contributions of rings in L40 and L160 are not apparent withinthe linear viscoelasticity.
Observation of stress and viscosity overshoots under biaxial elon-gational flows
The stress-strain curves σ B ( ε ) and the viscosity growth curves η B ( t ) under biaxial elonga-tional flows with the strain rate ˙ ε are represented in Fig. 2 on top and bottom, respectively.The Savitzky-Golay filter was used for smoothing the time series data of the biaxial elon-gational stress σ B = ( σ xx + σ yy ) / − σ zz and the biaxial elongational viscosity η B = σ B / ˙ ε .9igure 2: Stress-strain curves (top row) and viscosity growth (viscosity-time) curves (bottomrow) under biaxial elongational flows with biaxial elongation rates ˙ ε . Left, middle, and rightcolumns represent R160-L10, R160-L40, and R160-L160 blends, respectively. Each colorrepresents the corresponding elonation rate ˙ ε . Each graph shows lines with colors. Theselines were obtained from pure linear melts (L10, L40, and L160) for comparison amongring-linear blends (b) and pure melts of linear chains (p).10ymbols and lines shown in Fig. 2 are representing the melts of ring-linear blends and puremelts of linear chains, respectively.The stress-strain curves (top row in Fig. 2) show the strain hardening phenomena, whichis the nonlinear lift-up at a strain around unity, and the steady states for ε >
10 as same asthe entangled linear polymers. The viscosity growth curves (bottom row in Fig. 2) capturethe linear and nonlinear phenomena; the viscosity growth curves overlap with each other inthe linear regimes and deviate from a linear growth curve in the nonlinear regimes ˙ ε > /τ (for linear chains) or 1 /τ (for rings). As we expected from the relaxation modulus shown inFig. 1, R160-L10 show the strain hardening phenomena owing to rings because the two-steprelaxation in Fig. 1 comes from the component of rings, and the steady state viscosities inR160-L160 are weakly enhanced rather than the pure melt of L160.Against our expectations, R160-L40 exhibits the strain hardening phenomena owing torings and the viscosity overshoot when ˙ ε ≥ . /τ e ). The viscosity overshoot inR160-L40 is more pronounced in the higher strain rate. The strain hardening phenomenaobserved in R160-L40 are not so curious because the small amount of long rings has the longrelaxation time than the host linear chains, and is expected to cause the nonlinear behavioras same as the small amount of long linear chain. Our surprise is the viscosity overshootobserved in R160-L40. As R160-L10 do not show such overshoot phenomena, entanglementsor “hooking” among ring and linear chains are important to show the overshoot. In R160-L160, however, the overshoot is suppressed (or significantly small). Thus, this overshootphenomena are not observed in highly entangled melts.Moreover, focusing on the smallest values of strain rate ˙ ε s showing nonlinear strain hard-ening behaviors, we found a curious dependency; ˙ ε ≥ . ε ≥ . ε ≥ . ε s in R160-L10 and R160-L40 shouldbe exactly the same. Because, the longest relaxation times in R160-L10 and R160-L40 arethe same, corresponding to that in R160, as observed in Fig. 1. The smallest strain rate ˙ ε s should be determined by the relaxation time of R160 as ˙ ε s = 1 /τ , R160 . However, this con-ventional hypothesis is not correct in the ring-linear blends as found here. Thus, the shift ofthe smallest strain rate is coming from the cooperative phenomena of ring and linear chains.The dependency of the smallest strain rate can be estimated as ˙ ε s ( N L ) ∼ N − / accordingto our findings, ˙ ε s ( N L = 10) = 0 . ε s (40) = 0 . ε s (160) = 0 . ε s ∼ /τ , where therelaxation time τ depends on chain length N as τ ∼ N or N . for linear chains without orwith entanglements, respectively. We expect that the fluctuation of the linear chain affectsthe smallest value of strain rate ˙ ε s because of the power of N L , i.e. − /
2. Its origin is unclearat the present stage. Further discussion on ˙ ε s ( N L ) will be provided from our ongoing project.In the followings, we focus on R160-L40, which shows viscosity overshoot phenomena, andwould like to clarify its characteristics and the microscopic mechanism of viscosity overshootin the biaxial elongational flow. Direct observation of rings under biaxial elongational flow with ˙ ε = 0 . (R160-L40) Figure 3 shows snapshots of rings under biaxial elongational flow with ˙ ε = 0 .
001 in R160-L40. Each center of mass of ring is set to origin and all rings ( M R160 = 384) in R160-L40 aredisplayed. One of rings is highlighted for guide to eyes. We used OVITO for visualization.Focusing on the top and bottom rows in Fig. 3, the size of rings expands in the elonga-tional plane with increasing the strain ε , and the ring center opens around ε = 8, where theviscosity shows maximum. For ε >
8, the size of distribution in the elongational plane is12igure 3: Snapshots of rings under biaxial elongational flow with ˙ ε = 0 .
001 in R160-L40. Topview (seeing xy -plane, or the elongational plane) and side view ( xz -plane) are presented topand bottom, respectively. Each column corresponds S n ( n = 0 , , , , , , ε = n . Thick rings with red color are identical one. The graphs of radial distribution function(RDF) are presented on the middle row. 13lightly reduced, and the thickness of distribution along z -direction is increased with increas-ing the strain ε . Because the size of distribution in the elongational plane and the thicknessof distribution along z-direction contribute positive and negative to σ B , respectively, we canunderstand the viscosity overshoot observed in R160-L40 is caused by the dynamical behav-ior of rings under biaxial elongational flows. In small strains up to the maximum viscosity,the rings are gradually stretching in the elongational plane. After achieving the maximumviscosity, the rings stretched in the elongational plane are slightly relaxed. To identify theorigin of the relaxation of rings, we investigate the number of linear chains penetratingthrough rings as in the followings. Probability Distribution of number of linear chains penetratingthrough rings (R160-L40)
Figure 4: Probability distribution function of number of penetrations P ( n p ) under biaxialelongational flow with ˙ ε = 0 . n ( n = 0 , , , , , ,
30) represents ε = n . Each dashedline, prepared for guide to eyes, represents Weibull distribution obtained from a nonlinear-least-squares curve fitting. Inset shows n p at the peak of Weibull distribution against S n .Figure 4 shows probability distribution function of number of penetrations, P ( n p ), underbiaxial elongational flow with ˙ ε = 0 . f ( n p , λ, α ) = αλ (cid:0) n p λ (cid:1) α − exp (cid:0) − (cid:0) n p λ (cid:1) α (cid:1) , as a fitting function of P ( n p ) for guide to eyes. Tocount the number of linear chains penetrating through rings, we checked for all pairs of ring14nd linear chains, and judged whether the selected linear chain is penetrating through theselected ring or not. The method has been proposed in our recent work and explained asfollows. We make a loop from a linear chain connecting both ends. This extra connection isalways placed outside of the ring. If the ring and the loop made from the linear chain makea catenane, the linear chain is judged to be penetrating through the ring.As increasing the strain up to ε = 2, the peak of P ( n p ), where n p = λ (cid:0) − α (cid:1) α , shifts tolarger n p side. At the same time, the height of peak decreases with increasing the varianceof distribution. For 2 < ε <
8, the peak position of P ( n p ) goes down to n p = 0 as shownin the inset of Fig. 4. When ε ≥
8, the probability distribution of n p is suppressed inlarger n p side and increased in lower n p side with increasing the strain. The shape of thedistribution changes unimodal to downslope around the strain at the maximum viscosity, ε ≈
8. The causal relationship among the distribution of number of penetrations and theviscosity overshoot is still unclear. Further investigations are needed.
Comparisons with the other flow types (R160-L40)
Figure 5: Viscosity growth curves under typical three (uniaxial, planar, biaxial) elongationalflows, and shear flow in case of R160-L40. Left three graphs represent the cases for uniaxial,planar, and biaxial elongational flows in the order left to right. Most right graph shows thecase of shear flow. The black dashed lines represent the linear viscosity growth curves; 3 η ( t ),4 η ( t ), and 6 η ( t ) for the uniaxial, planar, and biaxial elongational flows, respectively, and η ( t ) for shear flows.Finally, we summarized the viscosity growth curves under uniaxial, planar, biaxial elon-gational flows and shear flows in case of R160-L40. The definitions of the viscosities, the15tresses, and the strain rate tensors for uniaxial, planar elongational flows and shear flowsare as follows; η U = σ U / ˙ ε , η P = σ P / ˙ ε , η S = σ S / ˙ γ , σ U = σ zz − ( σ xx + σ yy ) / σ P = σ zz − σ xx , σ S = σ xy , K = diag( − ˙ ε/ , − ˙ ε/ , ˙ ε ) for uniaxial elongational flow, K = diag( − ˙ ε, , ˙ ε ) forplanar elongational flow, K xy = ˙ γ (the other components are zero) for shear flow. Thelinear viscosity growth curve η ( t ) is obtained from the relaxation modulus G ( t ) such that η ( t ) = (cid:82) t G ( t (cid:48) ) dt (cid:48) . In the linear regimes, the viscosity growth curves correspond to the linearviscosity growth curves; 3 η ( t ), 4 η ( t ), and 6 η ( t ) for the uniaxial, planar, biaxial elongationalflows, respectively, and η ( t ) for shear flows. As found in the uniaxial and planar elongational flows, and the shear flow, the nonlinearbehaviors, strain hardening in elongational flow and shear thinning in shear flow, are observedowing to the rings longer than the host linear chains. The nonlinearity is, however, muchpronounced in the biaxial elongational flow. We can say that the cooperative behavior ofrings and linear chains is enhanced under biaxial elongational flow.
Summary
We investigated the ring-linear blends under biaxial elongational flows by coarse-grainedmolecular dynamics simulation. We found the cooperative phenomena of ring and linearchains in the nonlinear flow regimes. The viscosity overshoot phenomena were observed inthe R160:L40=1:10 blend (R160-L40) and pronounced in the higher strain rates. Moreover,we found the curios relationship that the chain-length dependency of the smallest values ofstrain rate showing the strain hardening was estimated as ˙ ε s ( N L ) ∼ N − / . To find out thecause of the stress overshoot, we directly observed the rings under biaxial elongational flowsand investigated the number of linear chains penetrating through rings. We found that thesize of rings in the elongational plane increased in small strain up to the maximum viscosity,and then the rings stretched in the elongational plane were slightly relaxed, correspondingto the viscosity overshoot. The relaxation of the stretched rings is related to the number of16enetrations. We concluded that the viscosity overshoot found in R160-L40 under biaxialelongational flow is owing to the maximum stretching in the elongational plane and therelaxation of stretched rings.The chain-length dependency of the smallest values of strain rate showing nonlinearity,˙ ε s ( N L ) ∼ N − / , can be clarified by usual experiment of biaxial elongation because it willbe observed where the strain is around one. There are no technical difficulties for suchsmall strain. The stress overshoot phenomena, however, are difficult to be observed in thepresent experimental techniques, which cannot achieve to strains over tens under biaxialelongation. Recent technological progress, such as the development of new rheometers, has been remarkable. Our numerical predictions of the cooperative phenomena among ringand linear chains are expected to be clarified in the future. Acknowledgement
TM would like to thank Prof. T. Taniguchi, Prof. M. Sugimoto, Prof. J.-I. Takimoto, Prof.S. K. Sukumaran, and Prof. T. Uneyama, for their fruitful discussions, comments, and en-couragements. The authors thank Prof. H. Jinnai, Prof. T. Sato, and Prof. T. Deguchi fortheir encouragements. For the computations in this work, the authors were partially sup-ported by the Supercomputer Center, the Institute for Solid State Physics, the Universityof Tokyo, the Center for Computational Materials Science, Institute for Materials Research,Tohoku University, the Joint Usage/Research Center for Interdisciplinary Large-scale In-formation Infrastructures (JHPCN) and the High-Performance Computing Infrastructure(HPCI) in Japan: hp200048, hp200168. This work was partially supported by JSPS KAK-ENHI, Japan, Grant Numbers: JP18H04494, JP19H00905, JP20K03875, JP20H04649, andJST CREST, Japan, Grant Numbers: JPMJCR1993, JPMJCR19T4.17 eferences (1) Doi, M.; Edwards, S. F.
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