Elastic behavior of spherical nanodroplets in head-on collision

5 Pages·0.23 MB·English

by  Sangrak Kim

Elastic behavior of spherical nanodroplets in head-on collision
∗
Sangrak Kim
Department of Physics,
Kyonggi University
94-6 Eui-dong, Youngtong-ku, Suwon 440-760, Korea
(Dated: January 28, 2010)
Simulation results for head-on collisions of equal-sized spherical polymer nanodroplets using
molecular dynamics are presented. Elastic behavior of an initial compressed phase for the col-
0 liding droplets is analyzed. Deformations and contact radii of thenanodroplets are compared with
1 the Hertzian model of elastic solid balls. It is found that at least the initial phase of collision can
0 beexplained bythis continuum model, except at the verymoment of the beginning of collision.
2

n
I. INTRODUCTION II. THEORY
a
J

8 When two elastic balls are colliding with each other,
2 conservationlawsforenergy,linearmomentumandangu-
A nanodroplet has so large a surface-to-volume ra-
larmomentumaregenerallyappliedbefore andafter the
] tio that its behavior is quite different from the ordi-
h collision. In this case,the collisionprocessis treatedlike
narymacroscopicdropletswhichweseeinevery-dayphe-
p a black box: we ignore the collision process itself. How-
- nomena. Recently, there are growing interests in nan- ever, we want to know what is going on during the colli-
p odroplets collisions[1]-[5], since they are closely related
sionprocess. Thiscollisionprocessseemstobesimilarto
m
withapplicationssuchasnanofluidics,drugdelivery,ink-
that of the statical compression of two balls, which can
o jet printing, etc. In particular, Deming and Mason[4]
be explained by the Hertz model. Here, we will summa-
c made nanodroplets for fighting cancer. Their eventual
rize the Hertzian model[7]-[8] of contacting elastic balls
.
s goalistomakepossiblethatthenanodropletscanharm- which have an equal-sized spherical shape with a radius
c lessly enter into cells and then release medicinal cargo.
i R. The contact is made by pushing them together. Fig.
s But making better drug-delivery vehicles still needs a
1a) shows the very moment of contact. Let us denote
y fundamental understanding of nanodroplets behavior. ′
h the coordinate system x and x as being positive in ei-
p ther direction from their contact point. Fig. 1b) shows
[ their squeezed state by a force F where F is a normal
Contactor collisionproblems in generalare very com- n n
component of an applied force from the other ball. The
1 plicatedtomodeltheoreticallyorevennumerically. Here,
deformedsurfacesarethus describedby equations ofthe
v we focus on the head-on collisions of two equal-sized
′ ′
2 spherical polymer nanodroplets using molecular dynam- form x = x(y,z) for the right ball and x = x(y,z) for
′
8 ics. Moleculardynamics(MD) hasbecome anindispens- the left ball, respectively, as in Fig. 1. Let u and u
0 be the x-components of the displacement vectors of the
able tool to study the phenomena in nano scale[6]. To
5 contact point on the deformed surfaces of the two balls,
understand the initial compressed phase of the collision,
.
1 deformations, longitudinal forces and contact radii, etc respectively. They are symmetricallydeformedaboutan
0 axisconnectingthecentersofballs. Then,fromFig. 1b),
are measured from the simulations. The simulation re-
0 we have the equality,
sultsareanalyzedwiththeHertzianmodel[7]-[8]ofcollid-
1
ing elastic solid balls. Originally, the Hertz model deals
: ′ ′
v with compressed elastic balls such as steel balls. It as- x+u+x +u =ξ, (1)
i
X sumes Hooke’s law between stress and strain. As far as
whereξisatotaldeformationofthecollidingballs. Thus,
I know, there is not yet any attempt to connect nan-
r
the total deformation ξ varies with the force F .
a odroplet collision with the Hertzian theory of elasticity. n
Thus this is a first report to explain the initial phase of The material is assumed to be isotropic and homoge-
droplet collision. neous. Linear elastic Hooke’s law holds between stress
and strain. It is further assumed that there is no ad-
hesion between balls. Furthermore, we consider only a
We will first present the Hertzian theory of elasticity small ξ limit. In this limit, the total deformation ξ is
for equal-sized spherical elastic balls. Next, we describe given by
MD simulation method and results. Then, these simu-
lation results are compared with the theoretical predic- ξ =F 2/3(γ22)1/3, (2)
tions. Finally, we will summarize our findings. n R
whereγ ≡3(1−ν2)/2Y, ν isaPoisson’sratioandY isa
Young’s modulus. The spherical shapes of balls become
∗E-mailaddress: [email protected] flat with the collision as shown in Fig. 1b). The contact
2
Z We performedMDsimulations byusing LAMMPS[11]
x u’ u’ x package with some modifications or additions, if neces-
sary. Visualizations of the simulation system were done
by using VMD[12]. Hereafter all the quantities will be
z expressed in the LJ reduced units.
We prepare two colliding spherical droplets as follows.
M M’
a ξ First, we make a polymer melt with chain length of
x x L = 10 such that their positions and velocities corre-
x’ O x
R R spond to the density of ρ = 0.5 and temperature of
T = 0.5. To make a spherical droplet, we connect each
R R
molecule with a virtual harmonic spring with a small
spring constant. The other end of each spring is con-
nected to the center of the simulation box to compress
a) b) thedroplettowardsitscenter. Thenwecanhaveaspher-
icaldropletwithradiusR=22.87withathermodynamic
FIG. 1: (Color on-line) a) Two equal-sized spherical elastic state of ρ = 0.83 and temperature of T = 0.5. After
balls with a radius R at the moment of contact. b) A com-
the shape of droplet becomes spherical, we turn off the
pressed state. Dotted lines represent the boundaries of the
connectionofharmonicspring. Thenumberofmolecules
two balls at the moment of their contact. They are com-
composingonedropletisn=40,000sothetotalnumber
pressedwithacontactradiusaandtotaldeformationξ. The
ofmoleculesin the systemis N =2n=80,000. Another
balls just begin to touch each other at the points M and M’,
colliding droplet partner is made with the configuration
respectively.
at different time in equilibration process. Thus they are
atthesamethermodynamicstatebutwithdifferentcon-
radius a of this flat contact circle is also given by figurations. Next, they are displaced an appropriate dis-
tance. Thenthetwodropletsareapproachingeachother
a=F 1/3(γR)1/3. (3) withthesamespeedvtocollide,sothattheirrelativeve-
n 2 locityisgn =2v. Wechooseasignificantlylargervalueof
cutofflengthr =5.0forLJinteraction,comparedtothe
c
usual value of r = 2.5, since, otherwise, they will expe-
c
III. SIMULATION RESULTS rience unwanted forces, when they enter the interaction
cutoff region from the far distance.
Now, we simulate head-on collisions of equal-sized Since the molecules in different droplets repel each
spherical polymer nanodroplets. To keep the spherical other, the droplets are eventually recoiled. We can mea-
shapeofthedroplets,wechoosepolymernanodroplets[9], sure their respective recoil velocities and thus the resti-
since a polymer nanodroplet can have less or no evapo- tutioncoefficienteisdetermined. Thecoefficientof(nor-
ration during the simulation. mal) restitution is given by
First, let us briefly describe the simulation method.
′
g
The molecular interactions are taken as follows. The e=− n, (6)
polymer chains are modeled by a rather abstract but gn
well-studied bead-spring model[10]. The chain length of ′
where g and g are normal components of relative ve-
a polymer is L =10. All monomers in the same droplet n n
locities of the colliding objects before and after collision,
interactwitheachotherthroughtheLennard-Jones(LJ)
respectively. Anyway,eisrelatedwiththeforcesF ,the
n
potential, given by material properties and the impact velocity g , etc. The
n
V (r)=4ǫ[(r/σ)−12−(r/σ)−6]. (4) simulationresultsaresummarizedinTableI. Noticethat
LJ a coefficientof restitutione is stronglydependent onthe
approaching velocity v, since colliding objects are liquid
Neighboring monomers in the same chain in addition in-
droplets,notthe solidclusters. The largertheapproach-
teractwiththefiniteextensionnon-linearelastic(FENE)
ing velocity v is, the smaller a coefficient of restitution e
potential,
becomesinconformancewiththeearlierresults[2]. Their
k recoil velocities reduce significantly, since their energy is
VFENE(r)=− Fr0log[1−(r/r0)2], (5) dissipatedin the collision. Thus their coefficient of resti-
2
tution e is quite small compared to the solid clusters.
wherekF isaspringconstantandr0isamaximumlength Now,letusdescribetheoverallcollisionprocesses. The
within which the chain can be maintained. In our sim- collision snapshots are shown in Fig. /refoverall at four
ulations, we choose r0 = 1.5 and kF = 30.0. Between different times at t = 0, 5.0, 27.5, and 75.0. Of course,
molecules in different droplets,only repulsiveterm of LJ the shapes of colliding droplets significantly change with
potential is applied so that the droplets repel each other time as expected. At t = 0.0, one droplet has roughly
to match with the Hertz model. a spherical shape with a radius R ≈ 22.09 and the sep-
3
TABLEI:Simulationdataofcoefficientsofrestitutionewith
7000
impact velocity v.
v 0.1 0.2 0.5 1.0 1.5 2.0 6000
e 0.437 0.349 0.259 0.174 0.131 0.105
5000
aration gap between two droplets is set at ∆x0 = 5.0. 4000
They areapproachingeachother with a relativevelocity Fn
2u =3.0. Thespheresareunilaterallycompressedalong 3000
o
the longitudinaldirectionupto t=6.5,as shownin Fig.
2000
/refoverallb). Note that the colliding region is squeezed
into a flat surface, but the rest still keeps a spherical
1000
shape. Then the droplet is elongated along the trans-
verse directions (in y-z plane) up to t = 30.0 as shown
0
in Fig. /refoverallc). At around t = 32.0, its shape is 0 2000 4000 6000 8000 10000 12000
Timestep
maximally deformed; it becomes roughly a prolate ellip-
soid with a length of minor axis L ≈18.15 and lengths
x
of major axis Ly ≈ 69.57 and Lz ≈ 71.22. Then follows FIG.2: (Coloron-line)OveralllongitudinalforceFx vs. time
t. Thedropletsareapproachingtoeachotherwithavelocity
a recoveryreturning to its starting shape up to t=55.0.
At t = 55.0, the droplets are finally separated and be- u0 =1.5. Notethesecond peak around at t = 11.0.
ginto recoileachother, i.e. they areeventuallyreceding
witharecoilvelocityv′ ≈0.14withsomesmallacquired
x
transverse velocities. Therefore, they are moving at an
odd angle from each other, even though the collision is
assumedtobehead-on. Duringtherecoil,theymaytend
to return to a spherical shape due to their surface ten-
sion. In other words, they are in the stress relaxation
with a long relaxation time.
Next, to examine the elastic behavior of the collision a) b)
process, we focus on the first part of initial compressed
phase. To see more detailed motion, we halved the size
of timestep during initial compressed phase, since the
particles involved during the collision are moving rather
rapidly in this phase. In Fig. 4, we present the force
F vs. deformation ξ in the initial compressed phase up
n
c) d)
to t = 5.0. The deformation ξ can be directly measured
with the standardimageanalysis. The dotted points are
FIG. 3: (Color on-line) Snapshots of collision progresses at
the simulation results. The solid line is fitted to the Eq.
differenttimes. a)t=0,separatedwithgapof∆x0=5.0be-
(2). This may come from two reasons: first, the very
tween two leading edges, approaching with a impact velocity
initial contact tip is not a continuous media. They are
2u =3.0, b) t = 5.0, initially compressing stage, still main-
o
composed of only a small number of molecules. Second, tainingasphericalshape,c)t=27.5,transverselyexpanding,
the shape of the tip is not spherical, but irregular and prolate ellipsoidal shape, d) t = 75.0, receding stage, in the
rough. The fitting is rather good except at small values process of returningto its spherical shape.
of F . From the fitting to the data using Eq. (2), we
n
obtain a value of
(2γ2/R)1/3 =0.0253. (7) asharpincreaseofstretchedS-shapeis observed. Butat
around a = 10.0, they show a linear increase. From the
Finally, to further confirm the validity of the theoret- fitting to the data using Eq. (3), we can again obtain a
ical prediction for the elastic theory of the initial com- value of
pressed phase of a droplet, the contact radius a in Eq.
(3) is calculated. The consequent results of the contact (γR/2)1/3 =0.679. (8)
radius a vs. force F are depicted in Fig. 5. The values
n
are also measured with the standard image analysis. In
the same time range, the deformation ξ vs. force F in From the Eqs. (7) and (8), we can get the estimated
n
Eq. 4 show the similar behavior as the prediction of Eq. value of R ≈ 36.45, which is rather larger than our
est
(3). In Fig. 5, when the radius a is smaller than 6.0, the initially setting value of R = 22.87, but the order of
force F is shapely increasing. Or, at around F = 800, magnitude is comparable.
n n
4
IV. CONCLUSION
10
9
8
7
6
ξ 5
4
3 Let us now conclude our results. Here, we mainly fo-
cused on the initial compressed phase for the head-on
2
collision of equal-sized spherical polymer nanodroplets.
1 In this phase, the phenomenon can be explained by an
elasticcompressedphaseinlongitudinaldirection. When
0
0 1000 2000 3000 4000 5000 6000 7000
the collidingregionistoocompressedtokeepthe spheri-
F
n
calshapeofthedroplet,itsshapechangesintoanprolate
ellipsoid, as shown in Fig. 3c). The next phase is thus
FIG.4: (Color on-line)Deformations ξ vs. longitudinal force
expansion in the transverse direction. After the com-
F in theinitial compressed phaseof a collision. Solid line is
anfittothedatabyEq. (2)withR2 =0.73and(γ22/R)1/3 = pression, energy exceeds the elastic limit to support the
sphericalshape,itmaythenbreakintoparts. Theelastic
0.0253. Notethe poor agreement near theorigin.
behaviors of the initial compressed phase of the collision
can be understood with the Hertzian theory of elastic
balls. The simulation results conform to this theoreti-
14 cal predictions. At least, the initial compressed phase of
a collision, the Hertzian theory of elastic balls explains
12 rather well the simulation results as measured with de-
formationsξ, longitudinalforcesF ,andcontactradiia,
n
10 except the very moment of collision. In this limit, the
continuum theory of the Hertz model is broken, since
8 there are very smallnumber ofmolecules involvedin the
a collisionandfurthermore,theshapeoftheinitialcontact
6 tip is far from spherical, but irregular and rough.
4
2
0
0 1000 2000 3000 4000 5000 6000 7000
F
n
FIG. 5: (Color on-line) Contact radius a vs. longitudinal More MD simulations should be followed at different
force F in the initial compressed phase of a collision. Solid thermodynamicstatesandsimulationcontrolparameters
n
line is a fit to the data by Eq. (3) with R2 = 0.78 and for further confirmations. We considered only the initial
(γR/2)1/3 =0.679. Note again the poor agreement near the compressedphase in this paper. To explain properly the
origin. later phases of the collisionprocess,we need alsoa more
elaborate theory of viscoplasticity or viscoelasticity.
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