The odd-even effect of the melting temperature of polymer film on finite lattice

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by  Tieyan Si

...
The odd-even effect of the melting temperature of polymer film on finite lattice
Tieyan Si
The Academy of Fundamental and Interdisciplinary Sciences,
Harbin Institute of Technology, Harbin, 150080, China
(Dated: October23, 2012)
Inordertofindaquantitativeunderstandingontheodd-eveneffectofmeltingfinitepolymers,we
computethemeltingtemperatureandentropygrowthrateofconfineddimerfilmonfiniterectangle
2 and torus. The theoretical melting temperaturedemonstrates similar behavioras theexperimental
1 observations. For an infinitely long dimer film belt with finite width, the melting temperature
0 strongly depends on the odd-evenness of the width. The entropy for an even number of width is
2 always larger than the entropy for an odd number of width. When the length of the rectangle
t goes to infinity, the speed of entropy growth shows a linear dependence on the width. This linear
c relationship holds both for rectangle and torus. Fusing two small rectangles with odd number of
O
lengthintoonebigrectanglegainsmoreentropythanfusingtwosmallrectangleswithevennumber
of length. Fusing two small toruses with even number of length into one big torus reduces entropy
8
insteadofincreasingit. Whilefusingtwosmalltoruseswithoddnumberoflengthwouldincreasethe
1
entropy. Theentropydifferencebetweencoveringtorusand coveringrectangle decaystozero when

thelatticesizebecomesinfinite. Thecorrelation functionbetweentwotopologically distinguishable
]
h loops on torus also demonstrate odd-eveneffect.
c
e PACSnumbers: 1.2.3.4.5.6
m
-
t I. INTRODUCTION to odd or vice versa. In the section III, we take an in-
a
finitely long dimer film with finite width, and study how
t
s the entropy growth rate behaves when we increase the
. Chemicalstudiesfoundthemeltingpointoffattyacids
t width. In section IV, we study how two small rectangles
a with odd number of chain length are below those with
m fuseintoonebigrectangle. Thesameprocedureforcom-
even number of chain length[1]. This odd-even effect
bining two small torus is computed to compare with the
- was clearly shown by the melting temperature of the n-
d caseofrectangle. InsectionV,thecorrelationfunctionof
n alkanes CnH2n+2[2]. When the length of the n-alkanes two loops on torus is computed to check its dependence
increases by one unit from an even number to its next
o on odd-even effect. In section VI, we proposed the gen-
c neighboring odd number, the melting temperature in- eral framework for calculating the melting temperature
[ creasesbyaverysmallvaluecomparedwithcaseforodd
of long polymers. The last section is a summary.
number growing to even number[2][3]. The odd-even ef-
1
fect is diminished when the chain becomes longer. The
v
6 miscibility of two different compounds also exhibit odd- II. THE MELTING TEMPERATURE OF DIMER
9 even effect. Two different compounds dissolve in each FILM ON FINITE RECTANGULAR LATTICE
7 otheronly whenthe lengthofthe linkerbetweenthem is WITH CONSTANT AREA
5 odd[4]. The odd-even effect is qualitatively understood
.
0 by considering how to pack large molecules with differ-
Themeltingtemperatureofpolymeristhecriticaltem-
1 ent structure. A quantitative understanding of odd-even
2 effect is still an open question, especially when it comes perature when a crystalline of polymers transforms into
1 a solid amorphous phase. Dimer is the simplest poly-
to anomalous odd-even effect[3].
: mer. We assume the crystalline phase of dimer film is
v
Dimer is the shortest polymer. Classical dimer model one frozen pattern of the dimers. Each dimer is locally
i
X counts the number of different ways to cover a graph by fixed in one direction. They can not rotate around. In
r pairing up two neighboring vertex[5]. The exact number the solid amorphous phase, the weak external bond con-
a of dimer covering on square lattice can be calculated by necting neighboring dimers is broken. The dimers are
Kasteleyn’s method[6]. The chemical polymer usually is free to rotate and reorient itself. But we can not make
longerthanadimer. Howeverthereisatheoreticalcorre- sure it must be in certain direction. So all different con-
spondence between dimer configuration and fluctuating
figurations are possible.
string configuration on two dimensional lattice[7]. Soft
We assume this phase transition is a first order tran-
longpolymer maybend into differentshapesdue to fluc-
sition, then the Gibbs free energy, ∆G=∆U T∆S, is
tuating environment. So we can get a basic theoretical −
zero at the phase transition point. We need to calculate
understanding on melting long polymer by studying the
howmuchthermalenergythefilmhasabsorbedtobreak
melting of dimer film.
the external bonds and the entropy of the two phases.
The article is organizedas following: In the sectionII, Supposethedimersarefrozeninthebeginning. When
we computed the melting temperature of dimer film on the dimers absorbed the energy of external heat source,
a constant area. The shape of the area varies from even it becomes active and oscillating around local equilib-
2
rium position. But the internal bond connecting two to width. We let n running from 2 to 16. The length of
monomersisstrongenoughtosurviveduringthemelting this rectangleatn=2 is (6.16287/2) 1018, itis approxi-
×
process. At very high temperature, dimer may also de- mately ten times longer than the length at n=16. When
compose into monomers, but that temperature is much the width growsfrom 2 to 16, the dimer film grows from
higher than the melting temperature. We denote the a long belt to a short belt with increased width. It can
energy quanta for breaking a frozen external bond as be viewed as folding an almost infinitely long polymer.
ǫ . The total thermal energy for melting the film is The melting temperature increases as the width grows
i
∆U = ǫ . We denote the partition function of the larger(Fig. 2). When the width grows larger than 16,
i i
frozen pPhase as Z1, and the partition function of the the melting temperature approaches to a constant num-
melted phase as Z2. Then the critical temperature can ber. Thenitreachesthethermaldynamiclimit,thefinite
be computed by size effect disappeared. The increase of melting temper-
ature at each step depends on the odd-evenness of the
∆U ǫ
T = = i i . (1) width. For example, when the width grows from 2 to 3,
c
∆S kBlog[Z2P] kBlog[Z1] the critical temperature increased by 0.175617 percent.
−
While it only increased 0.0561494 percent from 3 to 4.
We consider a finite rectangle of square lattice with m
When we observe the whole curve of ∆T , it shows the
rows and n columns. The total number of lattice sites is c
slop of ∆T from an even number to an odd number is
m n. Thetotalnumberofbondsis(2mn n m). The c
× − − always larger than its sequent neighboring case from an
total number of internal bonds within a dimer is mn/2.
Thetotalnumberofexternalbondsis 3mn n m. The odd number to an even number. This theoretical behav-
2 − − iorofmeltingtemperatureisquite similartothemelting
thermal energy would shake the external bonds, when
all of the external bonds become weak enough to reach temperature of n-alkanes CnH2n+2[2][3].
the threshold of collapsing point, the dime film melts.
75
The dimers randomlyrotateto formanew pattern. The
ttehenmeerrpmgeyaralotefunereaercgihysea∆xbUtseorr=nbael(d32bmboynndth−.eWndei−mtemark)efiǫl0imt. aǫas0tǫtis0he=thcer1i0tui−cn1ai6tl 70 æ æ æ æ æ æ æ æ æ æ æ æ
65
for the convenience of computation. æ
The entropyofthe frozenphase is zerosince the num- Tc
60 æ
ber of configurationis just one. In the melting phase we
have to count all possible dimer coveringson the rectan-
55
gle lattice. Kasteleyn had developed a method to count
allpossible waysforcoveringthe whole latticeby dimers æ
50
without isolated sites[6]. First one needs to construct
Kasteleyn’s adjacent matrix K by assigning each bond 2 4 6 8 10 12 14 16
of square lattice a number +1 or 1. The product of m
−
thesenumbersaroundanarbitraryrectanglemustbe 1.
−
The total number of all possible configurations of dimer
covering is Z = √detK [6]. Kasteleyn’s method was FIG. 2: The melting temperature of dimer film at different
equivalently mapped into quantum field theory of free width from 2 to 16.
fermion model [8]. A physical picture of dimer covering
is to place a fermion at each lattice site. Two neighbor-
ingfermionscanformapairindifferentdirection. There Ln æ
H
is a hard-core repulsive interaction between two dimers (cid:144)Tc0.15
L
since they do not overlap each other. L
n
H
Tc
- 0.10
L
1
+
n
HTc0.05 à
H æ
=
Tc à æ
D 0.00 à æ à æ à æ à æ à æ
2 4 6 8 10 12 14 16
FIG. 1: The dimer film is almost infinitely long . The width
varies from 2 to16. m
Wecomputedthemetingtemperatureofdimerfilmon
a large rectangle. The total number of covering lattice
FIG. 3: The increase of melting temperature at each step
sites is mn = Π7i=1i × 11×13 = 6.16287× 1018. We where thewidth of thedimer film grows bigger by one unit.
keep this number invariant but varies the ratio of height
3
200000 à à æ à æ à æ à
æ
à
æ
190000
à
æ
S180000
à
170000
FIG. 5: The width of the dimer film is finite and increasing
160000 stepbystep. Thelengthofthedimerfilmwouldbeinfinitely
æ
0 5 10 15 long.
m
1 for convenience. For fixed number of rows, we com-
pute how much entropy increased when the number of
FIG.4: Theentropyofdimercoveringson theinfinitelylong columns is increased by one unit,
rectangle. Thetotallatticesitesoftherectangleismn=Π7i=1i
×11×13=6.16287×1018. The width of rectangle is repre- ∆S =log[Z(m,n)] log[Z(m,n 1)]. (3)
− −
sented by m.
Fig. 6 shows the entropy increase on a rectangle and a
torus when the number of columns increases from 2 to
At the melting point, the dimers are not strongly con-
20. The number of rows is m = 10. When the length
fined in a local position. Some dimers maybe jump out
growsfrom odd number to even number, the step size of
of its home to another hole. This provides some proba-
entropy increase is much larger than the case of grow-
bility to reshape the soft film. We can assume the long
ing from even to odd(Fig. 6). As the number of length
rectangle can vary its own shape following the second
grows,theentropyincreasedecaysatevennumber,while
lawofthermodynamics. Whentherectanglereshapesit-
theamplitudeof∆S increasesatoddnumber. Thisphe-
self, it would exert some force to its surroundings. We
nomena holds both for rectangle and torus. Different
call this force entropy force. As the shortest length
boundary condition does not smear out the odd-even ef-
n = (6.16287/16) 1018 is almost infinitely larger than
× fect.
the longestwidth m=16,we view the lengthas infinity.
When the length becomes longer, both the two
We define the entropy force induced by increasing m as
branches of ∆S for even number and odd number con-
∆S(m,n) verge to the same limit value. It takes more steps
F =T , (2)
m for ∆S to converge on a torus than that on a rectan-
∆m
gle. The limit value ∆S(n ,m=10)on rectangleis
T is temperature. Here we take it as a positive constant →∞
∆S (n ,m=10) = 2.778441, here I have made a
rec
forsimplicity. Numericalcomputationshowstheentropy cut off at→10∞−6. The limit value at large n on a torus
at even number is larger than that at odd number(Fig.
is a little bit higher, ∆S (n ,m=10)=2.969359.
tor
4). If ∆m = 3 2, one can see ∆S(m,n) < 0 by direct →∞
− The difference between the two limit values of rectangle
observation of Fig. 4. So entropy force f(n,m(2 3))
→ and torus preserves until m . Fig. 7 showed the
is negative. For ∆m = 4 3, ∆S(m,n) > 0, the en- → ∞
− limit values of ∆Stor(n ) and ∆Srec(n ) when
tropy force f(n,m(3→4)) now becomes positive. This mgrowsfrom2to36. B→oth∞thetwolimitval→ues∞fallona
positive entropy force is much larger than the negative straightline. Forlargenumbersofm, m>100,thelimit
entropy force. If we enlarge the step size, ∆m=4 2=
− values can be fitted by empirical linear equations,
6 4= ,theentropyforceisconstantlypositive. Thus
− ···
the direction of entropy force depends on the unit scale. ∆S (n ,m) = 0.29m+C (m),
rec rec
Whenngrowslarger,theentropydifferencebetweeneven →∞
∆S (n ,m) = 0.29m+C (m),
tor tor
number and odd number vanished. So the entropy may →∞
m = 2k, k=0,1,2,3, . (4)
experience periodic decreasing in finite sized system be-
···
fore it reaches the maximal point.
Without losing the key features, here the accuracy of
thosenumericalvaluesinequationsarekeptto the order
of10−2forbrevity. Thisempiricallinearequationimplies
III. THE MELTING TEMPERATURE OF
DIMER FILM ON A GROWING RECTANGLE a constant,
∂ ∆S
We focus on the entropy increasing when the number κ= lim =0.29. (5)
∂m(cid:20)n→∞ ∆n(cid:21)
of columns increases one by one. The number of rows is
kept at a small value. Here the entropy of finite system This constant maybe has some hidden relationship with
isdefinedasthesameformulationofthermodynamicen- the entropy per site. However κ deviates far from the
tropy, S = k log[Z], Boltzmann constant k is set to exact number of the entropy per site in Ref. [6].
B B
4
L 4.5 à m ∆Srec(n→∞) ∆Stor(n→∞) ∆Stor−∆Srec
-1 10 2.778441055719985 2.969359257199864 0.190918
n 4.0 20 5.688333696422393 5.857553050376143 0.169219
H
S 30 8.601926338829257 8.764325608871156 0.162399
-
L 3.5 à 40 11.51650404237699 11.67554509633454 0.159041
n
H à 50 14.43148654142293 14.58852686413148 0.157040
=Sgle 23..50 æ à æ à æ à æ à æ à æ à æ à 120000 2598..0106832760066126383242049118 2598..1361143729785995708698695476 00..115531006973
an æ 300 87.31958020522484 87.47001658155840 0.150436
DSrect 2.0 æ 450000 111465..46735155682714507303466681 111465..67285164790859825182130490 00..115409190182
1.5
5 10 15 20
n TABLEI:Welisttentypicalvaluesoutofthewholedatabase.
(a)
à
L 4.5 Thedifferencebetween∆Stor(n )and∆Srec(n
-n14.0 à ∞Th)isisdiaffefruennccteioins roofotmed, i∆nCth(me t)op=→olCo∞gtoicra(mld)iff−erCenrecce(mb→e)-.
H
S
- 3.5 à tween a rectangle and a cylinder(since n , it is not
=HLSn3.0 à æ à æ à æ à æ à æ à ∆a Cto(r2u)s=0a.n4y00m16o2r.e).∆CT(hme)mdaecxaimysalwvhaelnuem→ofg∞r∆owCs(.mT)hies
æ most rapid decay occurs from m = 2 to m = 50. This
us2.5 æ
r suggest that the topological difference between a torus
o æ
St2.0 and rectangle is most obvious in finite scale. When m
D
continue to grow from m=50 to m=500, ∆C(m) only
æ
1.5 drops from 0.157040to 0.149912(TableI).
5 10 15 20
Thetopologicaldifferencebetweenrectangleandtorus
n
can be characterized by their different speed of entropy
(b) growth,
∆C(m)=∆S (n ) ∆S (n ). (6)
FIG.6: (a)Theentropygrowthofarectanglefromoddnum- tor →∞ − rec →∞
berofcolumnstoevennumberofcolumns,orviceverse. The
Number of rows is fixed, m=10. (b)The entropy growth of a The rapid decay of ∆C(m) is illustrated in Fig. 8. Nu-
torus. The number of rows is kept to m=10. The numberof merical check suggests that the decaying rate is much
columns varies from odd to even,or vice verse. faster than exponential decay. The topological differ-
ence between torus and rectangle will disappear when
the width goes to infinity. However ∆C(m) will have
finite non-zero value as long as m is finite. I call this
æà property of ∆C(m) as topological stability.
15 æàæà
æà
æà
L æàæàæà ¥L æ
D®¥HSn,m 1050 0æàæàæàæà1æà0æàæàæàæà2æà0æàæàæàæà3æà0æàæàæàæà40 50 ®¥-D®HLHSnSntorrec0000....11226802 æææææææææææææææææææææææææ
D
m 0 10 20 30 40 50
m
FIG. 7: The limit value of entropy growth for n → ∞ at
differentnumberofrowsm. Thesquaredotrepresentstorus. FIG. 8: The value of ∆C(m) = ∆Stor(n→∞) −
The discdot representsrectangle. Thelimit valueof torus is ∆Srec(n→∞) as a function of the number of rows m.
always higher than that of rectangle. ∆C(m) approaches zero as m goes to infinity.
5
IV. ENTROPY REDUCTION BY CUTTING æ æ
DIMER FILM ON FINITE RECTANGULAR 0.98 æ æ
æ æ
LATTICE AND TORUS 0.96 æ æ æ
gle0.94
n
a
Experiment observation found two different liquid Zrect00..9902 à à à à
crystal molecules only dissolve in each other when they D à à
0.88
are separated by odd number of spacers[4]. If we con-
sider the second law of thermodynamics, the fusion of 0.86
à à
two different types of molecule into one solution must 0 5 10 15
increase the entropy of the system. The larger amount p
of entropy increasing,the easierfor the two molecules to
(a)
dissolve. According to this maximal entropy principle, æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ
one can calculate the entropy before the two molecules 0
meet, Sa, and the entropy after they dissolve in each -2
odtehteerrm, Sinbe.s Twhetehevralutheeodfisesnotlrvoepywigllaionc,c∆urSor=nSotb.−TShae, orus -4 à à à à à à à
T
practical liquid crystal system is too complicate to ac- DZ -6 à à
complishanexactcalculationofentropy. I only focus on -8
theidealtheoryofdimermodeltoshowtheentropygain
-10
offusing two rectangleswiththeir length asodd number à à
is much larger than fusing two rectangles with a length 0 5 10 15 20 25 30 35
p
of even number.
We take a rectangle of square lattice with m rows and (b)
ncolumns,andkeepthenumberofrowsmasonebutdi-
videnumberofcolumnsnintotwonumbers: pandn p.
Then we calculate how much the number of dimer c−on- FIG. 9: (a) The increased number of states after dividing a
finiterectangle into two smaller parts. Thebig rectangle has
figurations increased when the two separated part fuse
m=18 rows and n=18 columns. m= 18 is invariant. The
into one,
length of n is divided into two parts, such as (1,17), (2,16),
Zm,n Zm,pZm,n−p (3,15), ···, and so forth. (b)The difference of total number
∆Z = − . (7) statesbeforeandafterdividingafinitetoruswithm=n=36.
Z
m,n
As ∆Z is not always positive, if we take the logarithm
function of ∆Z, negative ∆Z is excluded by the func-
tion. Entropy is a monotonic function of the num-
torus by keeping m invariant. The gap of entropy gain
ber of states. When the number of dimer configura-
between even case and odd case becomes smaller. The
tionsincreases(decrease),entropywillincrease(decrease).
entropygainoffusionatoddnumbersapproachestozero
Therefore we can use ∆Z to quantify the increase of en-
in mostcase except the casethat the cutting point is far
tropy in the following.
fromthemiddle. Intheothercase,Ikeepninvariant,but
I made a sequence of cutting one rectangles into two
doubled m. The entropy gain for fusing two odd torus
daughterrectangles, (1,n 1),(2,n 2),(3,n 3), .
{ − − − ···} becomeszero. Butentropyreductionfor fusing twoeven
The entropy gain for fusing two odd rectangles is much
torusbecomesstronger. Thegapofentropygainbetween
larger than that for fusing two even rectangles(Fig. 9
even and odd grows wider. Increasing n makes smaller
(a)). The entropy gain form two separated parabola
entropygap,whileincreasingmenlargertheentropygap.
band, the upper band is consist of all the divisions at
Therefore the two spatial dimension parameter play two
odd numbers, the lower band includes all the divisions
competing roles in controlling the entropy gap.
at even numbers. The same phenomena also occurs for
cutting one torus into two daughter toruses(Fig. 9 (b)).
Theentropygainforfusingtwooddtorusispositive. But In mind of the second law of thermodynamic, fusion
fusing two even torus does not increase entropy, instead oftwo rectanglespreferfusing twoodd rectanglesto fus-
the entropy after the fusion is reduced(Fig. 9 (b)). This ing twoevenrectangles. While for torus,only fusing two
is a significant difference from the case of rectangle lat- oddtorusesdoesnotviolatethesecondlawofthermody-
tice. The fusionoftworectanglesobey the secondlawof namics. In other words, a torus tend to divide into two
thermodynamics, no matter their length is odd or even. even toruses instead of two odd toruses. Even though
The fusion of two odd rectangle is favored by entropy. the oversimplified dimer modelis notdirectly relatedto
But for torus, only fusing two odd torus will result in theexperimentobservationofliquidcrystalmixture,one
increasing entropy. can still get a rough primary understanding on why two
Theentropygainforfusingtwoeventorusesisnegative polymers are miscible only when the linker includes odd
atalldifferentscales. Idoubledthe lengthofthe mother number of units.
6
0ææææææææææææææææææææææææææææææææææææææ angle. This entropy reduction quantifies the geometric
-2 æææææææææææææææææææææææ corIrneltahteiofnreoefftehremtiwono tlohoeposr.y of dimer model, the corre-
æ æ
-4
lation of two monomers is defined by the ratio of total
DZ -6 æ æ numberofconfigurationswithtwomonomertothatwith-
-8 out two monomers[9][10]. The two loops is a long string
-10 æ æ of many monomers. In most cases, the number of dimer
-12 configurations of rectangle is much smaller than that of
-14 torus. The subtle information of loop correlation lost in
0 10 20 30 40 50 60 70
zeros. ThusI usealogarithmfunctiontoextractoutthe
p
subtle information. The correlation of the two intersect-
(a)
ing loops is defined as
0ææ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ ææ
C(L1,L2) = log[ ψ1ψ2 ψnψoψ1ψ2 ψm ],
-20 æ æ æ h ··· ··· i
æ æ = log[ L1L2 ] (8)
h i
DZ -40 æ æ Loop L1 is a string of n+1 monomers, ψ1ψ2 ψnψo.
···
Loop L2 covers m monomers,ψ1ψ2 ψm, and the
-60 ···
monomersatthe intersectingpoint, ψ . This loopcorre-
æ æ o
lationfunctionquantifiesthecorrelationamong(n+m+
-80 0 5 10 15 20 25 30 35 1)fermions. Followingthemonomercorrelationinquan-
p tum field theory of dimer model, C(L1,L2) is essentially
(b) the entropy difference between rectangle and torus,
Z
rec
FIG. 10: (a) Thevariation of numberof states for dividinga C(L1,L2)=log[Z ]=Srec−Stor. (9)
tor
finite torus with (m = 72, n = 36) and uniting them. Com-
paringwiththefinitetorusinFig. 9(b),thenumberofrows When we keep total number of fermion as constant,
isdoubled. (b)Thedifferenceofnumberofstatesbefore and the total length of the two loops is constant, L1+L2 =
after dividing the finite torus with (m = 36, n = 72). The n+m+1 = const. The geometric correlation between
numberof columns is doubled comparing with Fig. 9 (b). the two loops can be understood by the shape of torus
andrectangle. Shrinking L1 will enlargeL2. If the circle
V. GEOMETRICAL AND TOPOLOGICAL L1 ofthetorusbecomesthinner,theothercircleL2 must
CORRELATION OF TWO INTERSECTING expands larger. For the rectangle, if L1 becomes longer,
LOOPS ON TORUS L2 grows shorter. Thus geometric correlation between
the two loops must be negative.
The negative correlation of two loops shows different
behavior according to even or odd number of fermion in
eachloop. Firstwekeepm+n=48andboth(m,n)even.
The correlation is weak at very small n and very large
n,themaximalcorrelationisaroundthe middle, n=24.
The opposite correlationbehaviorappears,whenwe still
keep m+n = 48, but varies n and m in a sequence of
oddnumber. Thestrongcorrelationnowappearsatvery
small n and very large n. Around n=23 is the minimal
correlation which is still larger than the maximal value
for the case of even number sequences. One conclusion
we can draw is the loop correlation for the case of odd
FIG. 11: (a) A torus with two topologically inequivalent cir-
cles, L1 and L2. (2) A torus covered by dimers can be con- number is much strongerthan the case of even numbers.
structedfromarectanglelatticebytakingperiodicboundary The correlation has a finite gap between odd and even
condition in two spatial dimensions. case. The gap is closed when the length of the two loops
grows to infinity.
From the point view of topology, a torus is a man- More over,we studied the correlationbehavior for the
ifold generated by sweeping one circle around another first hybrid case that m is even and n is odd. The cor-
circle(Fig. 11 (a)). A torus is equivalent to a rectangle relation behavior inherit half of the even case and half
with periodic boundary condition. The two circles, L1 of the odd case. When m+n = 49, m is even and n
andL2,aretheboundariesofrectangle(Fig. 11(b)). The is odd. The maximal correlation appear at small n, and
entropy of dimer configuration is greatly reduced when theminimalcorrelationappearsatlargen. Fortheother
the lattice manifold transforms from a torus into a rect- hybrid case that m is odd and n is even, m+n = 49,
7
the correlationcurveis the mirrorimageof the firstcase for L2 = 2k. Then we define the correlation difference
reflected by the axis n=23. function as the difference between a correlation function
withL2 =2k+1anditsneighboringcorrelationfunction
-15 æ with L2 =2k,
m −−− even
-16
æ n −−− even ∆C(L1,L2)=C(L1,L2 =2k+1) C(L1,L2 =2k)(.10)
-17 −
L2
L,L1-18 æ If we calculate the limit value of ∆C(L1,L2) for the
HC -19 æ æ length of L2 = n growing to infinity, it leads to almost
æ the same equation (6) in last section except a ( 1) sign,
-20 æ æ −
-21 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ lim ∆C(L1,L2)=∆Srec(n ) ∆Stor(n )(.11)
10 20 30 40 L2→∞ →∞ − →∞
n
(a) ThuslimL2→∞∆C(L1,L2)=∆C(m)isequivalenttothe
topologicaldifference function of torus and rectanglede-
-18 æ fined in last section. Here ∆C(m) can be explained as
-20 m −−− even æ æ the gradient of loop correlation,
-22 n −−− oæddæ æ æ æ æ æ æ æ æ æ æ æ æ lim lim ∆C(L1,L2)=0. (12)
LL2-24 æ æ L1→∞L2→∞
HCL,1-26 æ This gradient function ∆C(L1,L2) has finite nonzero
æ
-28 value as long as the length of the two loops are finite. It
is only when both the two loops become infinitely long,
-30
æ ∆C(L1,L2) will become zero.
10 20 30 40
n
(b) VI. THE MELTING TEMPERATURE OF LONG
-24 æ æ æ æ æ æ æ æ æ æ æ æ POLYMER’S FILM
æ æ
æ æ
-26 æ æ
L2
L
HCL,1-28 æ æ
m −−− odd
-30
n −−− odd
æ æ
10 20 30 40
n
(c)
FIG.12: (a)Thesumof thetwoloops isconstant,L1+L2 = FIG.13: Coveringofafinitesquarelatticebylongpolymers.
49, i.e., m+n=48. Both m and n are even number varying
The length of the polymer showed above is 3, 4, 5, 6. The
fmro+mn=249to. T48h.ele(bn)gtThhoefLto2tvaalrlieensgatshfoolflowthiengt,w(o2,4lo,6o,p··i·s,4580)., pboenlydmienrtoredpirffeesreennttsethleemne-natlakarynecsoCnfingHu2rna+ti2o.nSs.oftpolymercan
ThecorrespondinglengthofL2is(47,45,43,···,3,1). (c)The
sumofthetwoloopis49,m+n=48. Bothmandnvariesfrom
The melting temperature for a film of long polymers
odd numberto odd number.
can be computed by the same critical temperature Eq.
Geometriccorrelationstronglydepends onthescaleof (1) for dimers. First we need to countthe number of ex-
the two loops. While topological correlation is invariant ternal bonds. Then we count all possible ways to cover
when the two loops are expanding or shrinking. When the lattice by tri-mer(3-connected monomers), quadra-
the loop is expanding or shrinking, the total number of mer(4-connected monomers), pentago-mer(5-connected
fermions along the loop will increase or decrease. Thus monomers), and so on(Fig. 13). However it is a difficult
the correlation function for a fixed total number of par- mathematicalproblemtoexactlycountallpossiblepoly-
ticles along the two loops is not topological invariant. mer coveringsby long polymers. So far we only have the
However the difference between two selected correlation exactsolutionofdimercoveringproblem. Ifthe polymer
functions can be a topological invariant. We first fix the is rigid enough to keep the shape of a straight line, the
lengthofL1 =m,andcalculatethedifferenceoftwocor- entropy of longer polymer would be much smaller than
relationfunctionsoneofwhichhasoddnumberoflength the entropy of shorter polymers for covering the same
for L2 = 2k +1, the other has even number of length area. We can always cut the longer polymers shorter, as
8
a result, the possible configurations would increase. Ac- growth rate of dimer film on a finite rectangle with fi-
cording to Eq. (1), smaller entropy induces larger melt- nite number of length. When the rectangle grows from
ingtemperature. Althoughthenumberofexternalbonds an odd number of length to even number of length, the
also decreases for longer polymer, it falls far behind the entropy increases much more than that for the case of a
decreasing speed of entropy. Quantitatively we can pre- rectangle growing from an even number of length to an
dict that the melting temperature would increase when oddnumber oflength. This results holds for alldifferent
the length of the polymer increases. In mind of the ex- initial length of the rectangle or torus. When the length
perimentalobservationofodd-eveneffect, we believe the of rectangle or torus grows to infinity, the speed of en-
exact entropy of long polymer covering would demon- tropyincreasingshowslineardependenceonthewidthof
strate odd-even effect. In fact, it is possible to measure rectangle or torus. The speed of entropy increasing for
the melting temperature of a film in lab. So experiment a torus is always higher than that of a rectangle. The
maybeisanotherwayforsolvingmathematicalproblems. differencebetweenatorusandarectangledecaystozero
In reality, the n-alkanes CnH2n+2 is not very rigid for as the width grows to infinity.
large n. The soft chain would bend into different con- The correlationfunctionoftwoloops ontorus alsode-
figurations(Fig. 13). In that case, the entropy of longer pends on the odd-eveness of their length. We select two
polymermaybeislargerthanshorterpolymers. Thenwe topologically inequivalent loops on torus. The two loops
would meet the anomalous odd-even effect. For a more has one intersecting point and the total number of sites
complicatecase,ifwemixpolymersofdifferentlength,it covered by the two loops is kept as constant. Then we
is almost impossible to get exact mathematical counting enlargeorshrinkoneoftheloopsto oddnumberoreven
ofallpossiblecoverings. Butwecandividethe polymers number of length. The correlation for odd numbers is
into two classes: even-polymer and odd-polymers. The much stronger than that for even numbers.
odd-even effect would still play a role. We fuse two smaller rectangles into a bigger rectangle
and compute how much entropy increased after the fu-
sion. It suggest the entropy increasing for fusing two
VII. SUMMARY
small rectangles with odd numbed of length is much
largerthanthatforfusingtwosmallrectangleswitheven
Experiments observed that polymers with odd num- number of length. While fusing two small toruses with
ber of length has better miscibility than polymers with even number of length into one big torus experience an
even number of length. We believe mixing liquid crystal entropy decreasing. The entropy increases only for fus-
molecules of different length obeys the maximal entropy ing two small toruses with odd number of length. This
principle. Studyingthedimerfilmcangiveusabasicun- odd-eveneffect is consistentwith the secondlaw of ther-
derstandingontheodd-eveneffect. Sincealongpolymer modynamics. In a finite sized system, the entropy may
of even number of length can be decomposed into many experiencesmallamplitudeofdecreasingperiodicallybe-
dimers. While an odd number of length is composed of foreit finallyreachesthe maximalentropy. This entropy
many dimers and one monomer. decreasing comes from the confinement of environment
We calculate the melting temperature and entropy due to finite scales.
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