The Immunity of Polymer-Microemulsion Networks

0.47 MB·English

by  Guy Hed

The Immunity of Polymer-Microemulsion Networks
Guy Hed and S. A. Safran
Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel
Theconceptofnetworkimmunity,i.e.,therobustnessofthenetworkconnectivityafterarandom
deletion of edges or vertices, has been investigated in biological or communication networks. We
apply this concept to a self-assembling, physical network of microemulsion droplets connected by
6 telechelic polymers, where more than one polymer can connect a pair of droplets. The gel phase
0 of this system has higher immunity if it is more likely to survive (i.e., maintain a macroscopic,
0 connected component) when some of the polymers are randomly degraded. We consider the dis-
2 tribution p(σ) of the number of polymers between a pair of droplets, and show that gel immunity
n decreases as the variance of p(σ) increases. Repulsive interactions between the polymers decrease
a thevariance, while attractive interactions increase thevariance, and may result in a bimodal p(σ).
J

7
1 I. INTRODUCTION Examples of equilibrium, network-formingsystems in-
clude surfactantsolutions, gels of biologicalmolecules or

] synthetic polymers. A particularly elegant experimental
t The immunity of networks refers to the ability of a
f realization of a transient, self-assembling, physical net-
o network to preserve its function, even when a certain
workhasbeenreportedin[5]. Thesystemconsistsofoil-
s number ofedgesor verticesis removed. This attribute is
. in-water microemulsion droplets connected by telechelic
t of interest in many types of networks [1]. The immunity
a polymers; the latter have a hydrophilic backbone with a
ofgenetic,cellularnetworksreflectstherobustnessofthe
m hydrophobic group at each chain end. The insolubility
networkunder amutationordeletionofone ofthe genes
- [2]. The robustness of the metabolic network of single of these groups in the continuous water phase and their
d solubility in the oil droplets results, in some cases, in a
n cell organisms is demonstrated by their persistence and
networkofdropletsconnectedbythepolymers. Mixtures
o growth despite environmental interventions [3]. In com-
of telechelic polymers and microemulsions have a wide
c munication networks, immunity refers to the ability of
[ the network to pass messages among most of its vertices rangeoftechnologicalapplications,includingpaints,cos-
3 evenaftersomecommunicationlinesarecut,orsomever- metics and enhanced oil recovery. Precise control of the
structural and rheologicalproperties of such materials is
v tices are damaged [4]. A communication network is im-
essentialforeffectiveandreliableperformance. Suchcon-
7 mune to such an attack, if it retains a connected section
trolis possible using the telechelic additives,that forma
3 thatspansenoughverticestoallowsomecommunication
5 transient network with controlled rheological and struc-
(even if quite indirect) between any two vertices.
9 tural properties that depends only on the physical prop-
0 Inthispaper,weextendtheseideastoanalyzetheim- ertiesofthe systemsuchasthe size andconcentrationof
5 munity of a network composed of droplets connected by the polymers and the droplets [5].
0 polymers[5]. Thegelphaseofsuchasystemischaracter-
/ izedbyaconnectedpartofthenetworkthatspanstheen- Apart from its applied interest, the telechelic-
t
a tire volume. The immunity ofthe gelis characterizedby microemulsion mixtures serve as model systems for the
m the fraction of polymers that can be removed (by chem- understanding of a more general class of transient net-
- ical breaking, depolymerization or by shear forces) from works. The advantage of this particular system is that
d thenetworkwhilethegelmaintainsitsmacroscopicchar- the parameters that control the thermodynamics and
n
acteristics, and does not transform into the fluid phase. structure can be easily identified and independently
o
c That is, a gel is immune if the physical network that controlled: the concentration of possible vertices (the
: comprises the gel retains a macroscopically connected droplets) and the connectivity of the network (the num-
v
component even though a relatively high fraction of the ber of polymers per droplet).
i
X polymers that form the network have been deleted. The experimental systems [5] exhibit a phase transi-
r A simple polymer gel is a network of connected poly- tion from a single phase to coexisting high density and
a
mers in which eachedge (i.e., bond) of the network con- low density network phases. Zilman et al. [7] showed
sists of one polymer; in this case, the concept of immu- that this transition is expected from entropic considera-
nity is idential with that of percolation [6]. However, in tionsalone,andoccursevenwithoutanyparticularener-
thesystemofpolymersthatjoinmicroemulsiondroplets, getic interactionsbetweenthe droplets andpolymers. In
eachedge(i.e., bond)thatconnectstwooildropletsmay thispaper,weextendthe analysisof[7]andexaminethe
contain more than one polymer – and sometimes many immunityofsuchpolymergelstorandomdegradationof
polymers. The percolation problem deals with the re- the polymers such as those that can result from chemi-
moval of entire edges from the network, while immunity caldegradationorfromshearforces. Asthepolymersare
referstothe removalofindividualpolymersfromagiven removed,themacroscopicnetworkbecomesdisconnected
edgeinsystemswheretherecanbe manypolymersinan and the gel revertsto a liquid. We show the interactions
edge. betweenthe hydrophobicgroupsatthe ends ofthe poly-
2
mers have an important affect on the immunity and the is referred to as an edge of the network, whose vertices
phase behavior of the system. are the droplets.
The relation between the degradationof the polymers Fromtheapplicationsofpercolationtheorytogelation
and the breakdown of the network is not trivial. Each weknowthatifthefraction,κ,ofoccupiededgesexceeds
edge in the network may contain several polymers, each the percolation threshold, κ , of the lattice, the system
c
of which connect one pair of droplets. The fraction of is in the gel phase. In the usual discussions of gel con-
polymersthatmustbedegradedinorderforthenetwork nectivity and percolation, each edge is occupied by one
toloseitsconnectivitydependsonthedistribution,p(σ), polymer. After the degradation of a fraction q of the
of the number of polymers in an edge. As the variance polymers, the condition for the survival of the gel phase
of the distribution increases, the network becomes more is κ(1−q)>κ .
c
sensitivetopolymerdegradation,andwillloseitsconnec- However, the system we discuss here has two impor-
tivity when a smaller fraction of polymers is degraded. tant and interesting differences from the usual percola-
We consider the distribution, p(σ), for various physi- tion models:
cal situations. Since we are treating relatively rigid and
short polymers, we do not consider the interactions be- 1. The vertices of the polymer network only exist at
tween the chains, but focus on the interactions between those lattice sites at which the oil droplets are
hydrophobic groups at the end of the polymers, which present,soonlytwoneighboringverticesofthelat-
we denote as stickers. In particular, we predict the ef- tice thatareeachoccupiedby droplets canbe con-
fectofbothrepulsiveandattractiveinteractionsbetween nectedbyanedge. Inthefollowing,werefertosuch
the stickers within a droplet. Repulsive interactions are a pair as an edge even it is not connected by poly-
expected from entropic and excluded volume consider- mers,inwhichcaseweconsideritasadisconnected
ations, while attractive interactions can arise from Van edge. If a given droplet has a neighboring vacant
der Waals forces, or from the local change of curvature lattice site, we refer to the the droplet surface that
thatthe hydrophobicendsinduceonthe dropletsurface. face this site as a free face.
We find that repulsiveinteractions decreasethe variance
ofp(σ)whileattractiveinteractionincreasethevariance. 2. An edge may be occupied by a number, σ, of poly-
Attractive interactions between the stickers may lead mers, whose value is determined by volume lim-
to the co-existence of dense edges, with high density of itations (in the case of excluded volume interac-
polymers,withdiluteedges,withlowpolymerdensity. In tions), explicit interactions between the polymers
that case, the distribution of the edge occupation num- or between the stickers, and the polymer chemi-
ber, p(σ), is bimodal, and its width is of the order of its calpotential-whichisfixedbytheirconcentration
average. As the system approaches the critical point as- in the solution. The polymer chemical potential
sociated with the emergence of this two-phase behavior, determines the average number of polymers in an
the fluctuations in the number of polymers in an edge edge, hσi.
as well as the variance of p(σ), increase. Thus, as we in-
creasethestrengthoftheinteractionbetweenthestickers
towardits critical value, local concentration fluctuations III. NETWORK IMMUNITY AND EDGE
increase, and the immunity decreases. OCCUPATION DISTRIBUTION
In Sec. II we describe the underlying physical model
forthe networkformingsystemsofinterest,andthe sim-
In this section, we concentrate on the effect of the
plifiedtheoreticalmodelweuseinourcalculations. Next,
edgeoccupationdistribution,p(σ),ontheimmunity. We
inSec. III,weexaminesomesimpledistributionsforp(σ)
do not consider here the droplet-related degrees of free-
and demonstrate the dependence of the immunity of the
dom, but rather assume that they are fixed in a given
gel on the variance of the distribution. The effects of re-
(static) configuration. The following section considers
pulsive and attractive interactions between the polymer
the droplet-relateddegreesof freedom. The only degrees
ends on or in a droplet are presentedin Sec. IV. We use
of freedomwe relate to in this section are occupation, σ,
an expression for the Landau free energy of the system
of the edges between neighboring droplets.
of both interacting polymers and droplets to obtain the
Ifthedropletsaredenseenough,andmostoftheedges
effect of the interactions on the variance of p(σ), and we
areconnectedby one ormore polymers,then the system
predict the phase diagram of this system.
hasaconnectedcomponentthatspansits entirevolume,
and the system is in the gel phase. If the fraction of dis-
connected edges (i.e. those that contain no polymer), is
II. THE MODEL large, the system is in the fluid phase. A system in the
gelphasethatundergoesadegradationofafractionofits
Wemapthephysicalspacetoadiscreteregularlattice polymers, has higher immunity if fewer edges are com-
[7], where each site can be occupied by an oil droplet. pletely disconnected by the degradation process. This
Each pair of neighboring droplets may be connected by will depend on the number of polymers per edge and on
polymers. The set of polymers that connect such a pair the variation of this number throughout the network.
3
For example, if the system has a distribution of edges
inwhichsomeedgeshavelargenumbersofpolymersand
some have very few, it will have lower immunity than a
system in which the number of polymers in each edge is
the same. This is because the random removal of poly-
mersfromedgeswithlowerthanaverageoccupancymore
readily leads to the complete disconnectionof that edge,
as demonstrated in Fig. 1.
The probability for a given edge to be occupied by
σ polymers is p(σ); the fraction of unoccupied, discon-
nected,edgesisgivenbyp(0)-theprobabilityofapairof
neighboring droplets to have no polymers linking them.
Afterafractionq ofthepolymersisdegraded,thenew
occupation of an edge which had σ polymers is propor-
tional to the product of the probability of a number, σ′
ofpolymerstoremainintheedge,(1−q)σ′ andtheprob-
abilityoftheremainingpolymerstoberemoved,q(σ−σ′).
The resulting distribution can be written
FIG.1: Modelsofpolymer-microemulsionsystemswithdiffer-
∞
p˜(σ′)= p(σ) σ qσ−σ′(1−q)σ′ . (1) entdistributions,p(σ),ofthenumberofpolymersinanedge.
(cid:18)σ′(cid:19) We compare a homogeneous system (a) with 11 polymers in
σX=σ′ each edge (mean m=11 and distribution width s=0) vs. a
system with bimodal p(σ) (b), as given by Eq. 4 (m = 11,
The new fraction of unoccupied edges is p˜(0) =
s = 9). After a random deletion of a fraction, q = 0.85, of
p(σ)qσ. The higher this quantity for a given initial
σ the polymers, most of the edges in the homogeneous system
aPverage edge occupation, hσi, and a given degradation (c) survive, and it still retains a macroscopically connected
probability, q, the lower the immunity of the system.
component. Intheinhomogeneoussystem,mostofthedilute
edges are eliminated by the deletion (d) and the system no
longer has a macroscopically connected component.
A. Simple distributions
In order to elucidate relation between the variance of As the variance s increases, the system become less im-
the edge occupation distribution p(σ) and the immunity mune to random polymer degradation, as demonstrated
of the gel, we first study two simple distributions. The inFig. 1. Thisisrelevanttopolymer-microemulsionsys-
first is a uniform distribution, tems with a bimodal distribution of the number of poly-
mers in an edge, as expected when the attractive inter-
1 m−s≤σ ≤m+s
p(σ)= 2s+1 , (2) actions between the hydrophobicstickers atthe polymer
(cid:26) 0 otherwise
ends are large (see Sec. IV).
with an average m and variance S2 =(s2+s)/3. In the following sections we consider several physical
σ
models corresponding to different physical interactions
Ifwerandomlydeleteafractionqofthepolymers,then
between the polymers and stickers, that yield different
the fraction of unoccupied edges is
edge occupation distributions, p(σ). We calculate the
qm−s 1−q2s+1 variance for each situation to estimate the gel immunity
p˜(0)= · , (3)
2s+1 1−q and how it depends on the interactions.
which is an increasing function of s for any value of q.
Thus, in a distribution with a larger variance, the prob-
ability of an edge to remain connected is smaller, and it B. Non-interacting polymers
is less immune to random polymer degradation.
The second distribution we would like to consider is
Wefirstexamineanaive,butinstructive,modelwhere
the bimodal distribution depicted in Fig. 1:
any edge can accommodate an infinite number of poly-
mers. In order to compare it with other models, we ex-
1/2 σ =m±s
p(σ)= , (4) press the variance as a function of the polymer concen-
(cid:26) 0 otherwise
tration, which is the control parameter in most of the
with an average m and variance s2. After a deletion of experiments [5]. First, we calculate the distribution in
the grandcanonicalensemble, in which the chemical po-
afractionq ofthe polymers,the fractionofdisconnected
tential, µ , of the polymers is fixed. We then calculate
edges is p
the dependence of µ on the polymer concentration, to
p
p˜(0)=qmcosh(slogq). (5) determine µ .
p
4
The distribution is given by ensemble, where the number of polymers and droplets is
not fixed, and is determined by the chemical potentials
p(σ)=e−e−µpe−σµp , (6) µ and µ , respectively. We shall consider the possibil-
p d
σ!
ityofpolymersthatconnectneighboringdropletsaswell
where the factorial in the denominator comes from the as polymers that form loops, in which both sticker ends
factthat the polymers areindistinguishable. This distri- reside in the same droplet.
bution has an average and a variance hσi=Sσ2 =e−µp. Thephysicalmodelcanbeassociatedwiththevertices
(droplets)andedgesofanunderlyinggraph. Thevertices
are given by the sites of the underlying lattice which are
C. Excluded volume interactions
occupied by the droplets. The edges of the graph con-
nectpairsofneighboringoccupiedverticesonthelattice.
In a more realistic model, the excluded volume inter- Sinceeachsitemaybeoccupiedbyadroplet,thestatisti-
actionslimitthe maximumnumberofpolymersthatcan calensemble we considerhere includes the sub-graphsof
reside in a given droplet. There is thus a finite, maxi- thelattice(thoughmostofthemhavenegligibleweight).
mal number of polymers, σM, associated with one edge. We divide each droplet into faces, one for each edge
The volume restriction may arise from Helfrich-like, ex- of the graph that emerges from its vertex. If another
cluded volume interactions between the polymer chains, droplet occupies a vertex that is connected to the same
or from the finite area occupied by each sticker in or on opposite edge, then the corresponding face is associated
the surface of a droplet. with the edge between the two droplets, and we call it
If the only interactions between the polymers and be- a linked face. If there is no neighboring droplet at the
tween the stickers are short range excluded volume in- vertex connected to a given edge, we term it a free face.
teractions, we can assume that there are σ sites in an
M Weconsidersystemsinwhichtheradiusofgyrationof
edge, that can each be occupied by one polymer. The
a polymer is smaller than the radius of a droplet. Thus,
distribution of the edge occupation is
we treat each edge and each free face as a separate sys-
σ tem. In an experiment, where the total number of poly-
p(σ)=Z−1 M e−σµp , (7)
mers is fixed, the edges and faces interact through this
(cid:18) σ (cid:19)
constraint. In the multi-canonical ensemble, however,
where Z =(1+e−µp)σM is the partition function. theydonotinteract. Wecanthustreateachedgeorfree
Thefreeenergypersite(wetakeallenergiesinunitsof face as an independent system and write its partition
kBT)isfe =−log(Z)/σM. Theaverageedgeoccupation function explicitly.
is
As discussed above, excluded volume interactions im-
hσi=σ ∂fe = σM , (8) ply that a face can contain a maximum of σM stickers.
M∂µp 1+eµp This means that an edge has a maximal occupation of
σ polymers. These polymers can be links - with the
and the variance of the distribution p(σ) is M
twotelechelicstickersofa givenpolymerattachedto the
∂2f σ hσi two opposing faces of adjacent droplets, or loops - with
S2 =−σ e = M =hσi 1− .
σ M∂µp2 4cosh2(µp/2) (cid:18) σM(cid:19) both stickers of a given polymers attached to the same
face of a single droplet. A face can contain a maximum
(9)
of σ /2 such loops.
The variance here is smaller than in the case of non- M
interacting polymers (Eq. 6). In the non-interacting
case, the variance was equal to hσi, where in the present
caseitissmallerbyafactorof1−hσi/σ . Asexpected, A. Polymer free energy
M
the difference between the distributions vanishes for the
case σM ≫ hσi. The narrowing of the distribution by We consideramodelinwhich,exceptforthe excluded
the excluded volume interactions increases as the aver- volume interactions that determine the maximal num-
age number of polymers per edge approaches σM, where ber of polymers per droplet, σM, polymers interact only
the variance vanishes and the distribution approaches a through their stickers (telechelic ends). The interactions
delta function as σ → σM. This suggests that repulsive between the polymer chains have a negligible contribu-
interactionsbetweenthepolymersorthestickersincrease tion for the case of relatively short and stiff chains that
the immunity of the gel under a random degradation of we consider here. Thus the energy difference between
the polymers. an edge that has two loops on the opposite faces of two
neighboringdroplets,andanedgeinwhichthesefacesare
linked by two polymers, is simply 2e −2e , where
loop link
IV. INTERACTING STICKERS e istheconfigurationalfreeenergyofaloop,ande
loop link
is that of a link. We thus define an effective thermody-
We now calculate the free energy of a system of namic potential of a polymer in an edge
droplets and polymers in which the polymers show at-
tractive or repulsive interactions in the multi-canonical µ=µ −log(e−eloop +e−elink). (10)
p
5
An edge where all the polymers have a thermodynamic
potential µ is equivalent to a one in which loops have a
potentialofµ +e andlinksathermodynamicpoten-
p loop
tialofµ +e ,inthe sense thatbothsystemshavethe
p link
same partition function and hence yield the same distri-
bution p(σ). Thus, we assume that all polymers in the
edgehaveathermodynamicpotentialofµ,whetherthey
are loops or links.
In a free face, where the polymers are all in loop con-
figurations,thethermodynamicpotentialofapolymeris
µp+eloop =µ+∆, where ∆=log(eeloop−elink +1).
The free energy of an edge or a free face depends only
ontheeffectivethermodynamicpotentialofthepolymers
andthe interactionbetweenthe stickers,whichissimilar
in both the linked faces and free faces. We denote the
free energy of an edge as σ f (µ), where f (µ) is the
M e e
free energy density (free energy per polymer site) of the
face surface, or the free energy of one of the σ sites
M
FIG. 2: The variance of the number of polymers in a bond,
on the face. Since in our model a free face differs from
asafunction oftheaveragebondoccupation (determinedby
an edge only in the number of sites available (i.e., only
the chemical potential µ), for σM = 20. The values of the
σ /2polymerscanbeaccommodatedonafreeface)and
M attractive interaction, α, are in units of kBT.
in the effective thermodynamic potential, its free energy
is (σ /2)f (µ+∆).
M e
One can readily derive the free energy, f , of a system
e
with only excluded volume interactions: f = −log(1+
e
e−µ). We considered such a system in Subsection IIIC, This expression applies whether α is positive (attractive
where we ignored the configurational free energy of the interactions), negative (repulsive interactions), or zero.
polymers. Eq. 14 suggests a divergence of the variance for α ≥ 4.
The interactions between the stickers are accounted In a macroscopic system (σ → ∞), this signifies an
M
for within a mean field approximation. The total inter- instability related to macroscopic phase separation. We
action energy for an edge with σ polymers is written donotexpectsuchbehaviorforfiniteσ (whichismore
M M
as−ασ /2. The number ofstickersthatoccupy agiven realistic),butwedoobserveapronouncedincreaseinthe
M
edge is σ. We therefore write the interaction free energy variance,astheinteractionstrengthαincreases(seeFig.
due to two-body interactions among the stickers using a 2).
virial-typeterm: −(α/2σ )σ2. Thepartitionfunctionis
M
The increase in the variance is due to a bimodal dis-
σM σ α tributionofthe polymerdensityinthebonds,asdemon-
Z = M exp −µσ+ σ2
strated in Fig. 3. For large values of α, the distribution
(cid:18) σ (cid:19) (cid:18) 2σ (cid:19)
σX=0 M p(σ) resembles the bimodal toy distribution in Eq. 4.
= σM ∞ dφ e−φ2/2α eφ−µ+1 σM (11)
r2παZ−∞ h (cid:0) (cid:1)i enoInugshu,cohneamsyasyteomb,sewrvheentwtohkeinindtseroafcetdiogness caornensetcrtoinngg
We consider this expression for Z as the parti- the droplets: dilute edges containing one or a few poly-
tion function of an effective Hamiltonian, H(φ) = mers, and dense edges, containing about σ polymers.
M
σ 1 φ2−log(eφ−µ+1) , that has a minimum at φ , Both kinds induce the same effective attraction between
M 2α 0
give(cid:2)n by: (cid:3) thedroplets(seebelow). Underdegradation,mostofthe
dilute edgeswillbreakdown,andthe connectivityofthe
α
φ = . (12) network may decrease dramatically, as depicted in Fig.
0 eµ−φ0 +1 1d. If the system was in the gel phase, and the connec-
For σ ≫1 the distribution of φ tends to a sharp peak tivity is reducedbelow the percolationthreshold, the gel
M
aroundφ , andwecanusethe meanfieldapproximation will transform into a fluid phase.
0
hf(φ)i ≈ f(φ ) to estimate the average fraction ϕ¯ =
φ 0
We conclude by obtaining an analytical expression for
hσi/σ of sites occupied in an edge and its variance:
M
the mean-field free energy of an edge, which is a good
∂fe 1 ∂H(φ) φ0 approximation for the case σM ≫ 1. Expansion of the
ϕ¯=− = ≈ , (13) effectiveHamiltonianarounditsminimumyieldsH(φ)≈
∂µ σ (cid:28) ∂µ (cid:29) α
M φ H(φ )+ 1H′′(φ )(φ−φ )2, where H′′(φ) = ∂2H/∂φ2.
S2 = ∂2fe ≈ 1 −α −1. (14) Plug0ging 2this ap0proximat0ion into the integral of Eq. 11
ϕ¯ ∂µ2 (cid:18)ϕ¯(1−ϕ¯) (cid:19) andtaking the logarithm,we obtainthe grand-canonical
6
Each edge has a free energy, σ f (µ), and every free
M e
face has a free energy, (σ /2)f (µ+∆). The partition
M e
function of the system containing both polymers (which
have both interactions and configurational entropy) and
droplets (which have translational entropy) is given by
the sum over all droplet configurations:
σ
M
Z = exp N µ +N σ f (µ)+N f (µ+∆)
d d E M e F e
2
X h i
{s}
ζ
= exp N µ + σ f (µ+∆) +N J(µ,∆) ,
d d M e E
(cid:20) (cid:18) 2 (cid:19) (cid:21)
X
{s}
(16)
where µ is the chemical potential of the droplets,
d
J(µ,∆) = f (µ)−f (µ+∆) is the effective interaction
e e
between the droplets, mediated by the polymers, and ζ
is the number of nearest neighbors at each site. This is
equivalenttothepartitionfunctionofalatticegasmodel
[7].
We use a Landau type, coarse grained approximation
forthefreeenergy,asafunctionofthedimensionlesscon-
centration (volume fraction) of droplets, c = hsi, which
is the average occupation per vertex:
f =clogc+(1−c)log(1−c)
1 1
+ ζσ f (µ+∆)c+ ζσ J(µ,∆)c2 . (17)
M e M
2 2
ThefirsttwotermsinEq. 17givethelatticegasentropy;
the third term is obtained using N /V = c, where V is
d
the volume of the system; the fourth term is obtained
fromEq. 16usingthemean-fieldapproximationN /V ≈
E
FIG. 3: The distribution P(σ) of the number of polymers in ζc2/2. The dependence off onthe potential µd is linear
a bond, for σM = 20 and σM = 200. The effective chemical in the droplet concentration, and this term has no effect
potential µ is such that hσi= σM/2. The values of α are in on the phase behavior of the system.
unitsof kBT. In the coexistence region, there is an equilibrium of
droplet chemical potential, µ = ∂f/∂c, and of the os-
d
motic pressure, π = f − µ c, between the coexisting
potential for a single edge: d d
phases. In equilibrium, the polymer chemical potential
1 1 α must be the same in both phases; this is automatically
f (µ)=−log(Z)≈ H(φ )− log H′′(φ )
e σ (cid:20) 0 2 (cid:18)σ 0 (cid:19)(cid:21) ensured by working at constant µ. By solving these
M M
equalities we obtain the binodal curve in c×µ (droplet
α 1
= ϕ¯2+log(1−ϕ¯)− log(1−αϕ¯(1−ϕ¯)), (15) concentration-polymer thermodynamic potential) plane.
2 2σ
M Since in experiment it is easier to control the polymer
where ϕ¯ is determined by the value of µ, and the last concentration ϕ than the polymer chemical potential, it
equality is obtained using Equations 12 and 13. is usefulto predictthe phasediagraminthe c×ϕplane.
We perform this transformation using the identity
df
B. Droplet free energy ϕ= =σ f′(µ+∆)c+J′(µ,∆)c2 , (18)
dµ M e
(cid:2) (cid:3)
Each vertex of the network in our model can oc-
where the prime indicates a derivative with respect to
cupy one oil droplet. We define si = 0,1 as the oc- µ. This fixes the polymer chemical potential, µ, as a
cupation number associated with vertex i in the lat-
function of the number of polymers, ϕ. In the phase-
tice. The number of edges in a given configuration of coexistence region, above the binodal curve, the condi-
the system is N = s s , where the sum is over
E hiji i j tions of constant droplet chemical potential, µd, and os-
pairs of neighboring sPites. The number of free faces is motic pressure,π , for fixed polymer chemicalpotential,
d
NF = hijisi(1−sj)+sj(1−si). Thenumberofdroplets µ, yield values of the droplet density, c, for the coexist-
is N =P s . ing dilute and dense phases respecitvely. From Eq. 18
d i i
P
7
polymer concentration. In fact, in Fig. 4 the curve for
α=10 is almost parallel to the tie-lines.
Wenotethatdiluteedges(withoneorafewpolymers)
can also exist in the dense droplet phase, since they in-
ducethesameeffectiveinteraction,J(µ,∆),betweenthe
droplets as the dense edges, andtheir polymers have the
same thermodynamic potential, µ, as the polymers in
thedenseedges. Thisscenariomaychangeifweconsider
interactions between the edges.
V. DISCUSSION
In this paper, we have predicted the immunity of the
gel phase of a polymer-microemulsionsystem to random
degradationof polymers. The gel in this system consists
of oil droplets connected by telechelic polymers. The set
of polymers that link a given pair of droplets is defined
FIG. 4: The phase coexistence curves of systems with differ-
as an edge (i.e., bond) of the gel network. We consider
ent values for the interaction between the stickers, α (given
the distribution, p(σ), of the number of polymers in one
inunitsofkBT). Thecurvesareobtainedfromthecondition
of equilibrium of thedroplet chemical potentials, µd, and os- suchedge,andarguethatimmunityofthegelisinversely
motic pressure, πd, in the coexisting phases, using the Lan- related to the variance, Sσ2, of this distribution.
daufreeenergyisgivenbyEq. 17. Thex-axisrepresentsthe Repulsive interactions between the stickers (the
dropletvolumefraction,c,andthey-axisrepresentsthenum- telechelic ends of the polymers) reduce the variance S2,
σ
ber of polymers per available site, ϕ¯. The parameters used thus contributing to the gel immunity. Attractive inter-
are ∆=0.3kBT, ζ =4 and σM =20. action, which increase S2, reduce the immunity of the
σ
gel.
Attractive interactions may give rise to a bimodal dis-
this implies that each phase has a different value of the
tribution of the edge occupation, which can give rise to
polymer concentration, ϕ.
the co-existence of dense and dilute edges in the same
The critical point in the c×µ (droplet concentration-
macroscopicsample. Adiluteedgecontainsasmallnum-
polymer concentration) plane occurs at c = 1/2, with µ
ber of polymers, and is likely to vanish under a random
given by
degradationofthepolymers. Whenafinitefraction(say,
−4 1/2)of the edges are dilute, the fraction of edges surviv-
J(µ,∆)= . (19)
ζσ ing a degradation will be lower, and the system is more
M
likely to transform from the gel to the fluid phase.
The binodal curve of the systemis presented in Fig. 4
This reasoning is particularly applicable to the case
for different values of the edge interaction α. The max-
of a dense phase that coexists with a dilute phase (as-
imal number of stickers that can reside in one droplet
suming the dilute phase is not a gel). Since the polymer
is ζσM. Thus, the number of stickers per unit volume is density within the dense droplet phase is high, we ex-
cζσ ,andthemaximalnumberofpolymersperunitvol-
M pect– ifthe attractiveinteractionsarelargeenough–to
ume is cζσ /2 (assuming a prohibitively high energetic
M findabimodaldistributionoftheedgeoccupationwithin
cost of a hydrophobic sticker in the water). The number
the dense droplet phase. In that case, we are interested
of polymers per available site in the system is then
in the immunity of the dense phase, and therefore must
takeintoaccountthedistribution,p(σ),withinthedense
2ϕ
ϕ¯= . (20) droplet phase.
ζσ c
M
To test our predictions for the immunity of a system
Aphaseco-existenceregionispresentinasystemwith- withattractiveinteractionsamongthepolymers,wesug-
outexplicitstickerinteractions(α=0)[7]. However,the gestcomparingtheelasticstability(gel-likenature)with
interactions can markedly affect the shape of the coex- respect to polymer degradation of two systems (both in
istence curve and of the concentrations in the two co- thedensephase): onewithnostickerinteractions(α=0)
existing phases. As the interaction, α, increases, the ef- and one with strong interactions (say, α = 10). Accord-
fective interaction between the droplets, induced by the ing to our predictions, the system with no interactions
edges, increasesand becomes more sensitive to the poly- will be characterizedby edges each of which has approx-
mer concentration. As a result, the onset of phase sepa- imately the same number of polymers, while the system
rationissharper,andcharacterizedbyaphasewithvery withstronginteractionswillshowabimodaldistribution
dilute droplets and low polymer concentration that co- for the edge occupancy. We predict that after polymer
exists with a phase with high droplet density and high degradationthe fractionof survivingedgeswillbe larger
8
in the non-interacting system (see Fig. 1), and that this cal methods might be used to cause these molecules to
system is more likely to remain in the gel phase. change their conformation and depolymerize. The in-
Inordertoinvestigatetheimmunityofsuchgelsexper- tensity and duration of the optical beam can be used to
imentally, it is necessary to devise a method to degrade control the fraction of polymers broken.
the polymers at random and in a homogeneous manner
throughoutthe system. It may be difficult to do this us-
ing molecules in a solvent, since a solvent added to the
systemwillinitiallyhaveamuchhigherconcentrationat
Acknowledgments
thesurfaceofthegel(penetratingthebulkbyslowdiffu-
sion); thus the polymers near the surface are much more
likely to be degraded. TheauthorsthankL.Chai,M.Gottlieb,S.Havelin,J.
One way to achieve random degradation may be to Klein,C.Ligoure,G.Porte,andA.Zilmanforusefuldis-
use polymers that have been synthesized with a photo- cussions. This work has been supported by the German
sensitive molecule in the middle of each chain. Opti- Israeli Foundation and by an EU Network Grant.
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