Phase transitions in two-dimensional colloidal particles...

Phase transitions in two-dimensional colloidal particlesat oil/water interfaces

Bo-Jiun Lin and Li-Jen Chena�

Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan

�Received 2 October 2006; accepted 16 November 2006; published online 18 January 2007�

Enhanced digital video microscopy is applied to study the equilibrium structure of atwo-dimensional charged sulfate-polystyrene particle �2 �m in diameter� monolayer at decane/water interfaces. When the surface density is decreased, a sequential phase transition, pure solidphase→pure hexatic phase→ liquid-hexatic-coexisting phase→pure liquid phase, is observed. Inaddition, the transition between liquid and hexatic phases is first order, while the solid-hexatic phasetransition is second order. The temperature effect on this two-dimensional melting transition isdiscussed by performing the experiments at three different temperatures. The Voronoi �J. ReineAngew. Math. 134, 198 �1908�� construction is applied to analyze the defect structure in thetwo-dimensional particle monolayer. The pair interaction potential of the two-dimensional colloidalparticles is found to be a very long range repulsion and to decay with distance to the power of −3.© 2007 American Institute of Physics. �DOI: 10.1063/1.2409677�

INTRODUCTION

It has been a wide interest in the behavior of particlemonolayer at a fluid/liquid interface for several decades.1–11

In 1968, Sheppard and Tcheurekdjian1 estimated the strengthof steric barriers between stabilized spherical polymer par-ticles at the air/water interface from the measurement of iso-therms �surface pressure versus surface area� for particlemonolayers. Aveyard et al.2–4 explored the capability of par-ticles at interfaces to modify the stability of foams and emul-sions. The aggregation of particles at the air/water interfaceas well as the aggregation morphology had also beenreported.7–9 Due to the very long range repulsive forces be-tween colloidal particles at oil/water interfaces, the chargedparticle system can form a highly ordered structure under thecondition of a dilute density. The colloid-colloid interactionwas experimentally determined by a laser tweezers method.10

According to the Kosterlitz-Thouless-Halperin-Nelson-Young �abbreviated by KTHNY hereafter� theory,12–16 themelting process is mediated by topological defects in a two-dimensional solid. The KTHNY theory predicts that there isa sequential phase transition in the two-dimensional melting.That is, a transition from solid to hexatic phase and thenfollowed by the other transition from hexatic to liquid phase.A special kind of order, called “bond orientational order,” isused to characterize these phases in the KTHNY theory.

Translational order g�r� and bond orientation order g6�r�are used to characterize liquid phase, hexatic phase, and solidphase in the KTHNY theory. For the liquid phase, both g�r�and g6�r� are short-range order and decay exponentially. Forthe hexatic phase, g�r� is short-range order and decays expo-nentially, while g6�r� is quasi-long-range order and decaysalgebraically. For the solid phase, g�r� is quasi-long-rangeorder and decay algebraically while g6�r� is long-range order.

When bond orientational order is short range, it makes thestructure factor an isotropic ring pattern. Quasi-long-rangebond orientational order in the hexatic phase makes thestructure factor becomes a sixfold pattern and long-rangebond orientation order in the solid phase makes the diffrac-tion pattern to be six clear spots.

Melting phenomenon in two-dimensional systems wasexamined by using many experimental systems, which canbe classified into four categories:17 �1� layered liquid crystalphases, �2� electrons on the surface of liquid helium, �3�gases absorbed on graphite, and �4� a colloidal suspension ofcharged particles. X-ray measurement is usually applied tothe former first three systems that can only provide indirectinformation of two-dimensional systems from the diffractionresult. In 1987, Murray and Van Winkle18 first confined col-loidal particles between two glass slides to study meltingphenomena in two dimensions. The movement of colloidalparticles can be observed directly and recorded by digitalvideo microscopy method, and the structure factor, radialdistribution function, and bond orientational function canthen be evaluated by image analysis. The system has beenfrequently utilized to study phase transition phenomena intwo dimensions. In addition, several kinds of colloidal par-ticle have been applied to the system.18–23 Nevertheless thereare some restrictions. The system confines colloidal particlesbetween two glass slides. The gap of separation between twoslides is about 1.2 times particle diameter. If the gap of sepa-ration is smaller than �1.2 times particle diameter, the col-loidal particles become immobile. Because of this limitation,the particle can still move along the vertical directionslightly. Therefore, this type of system is usually called the“quasi-two-dimensional system.” Since the separation is al-ways slightly larger than the particle diameter, some three-dimensional effect may occur. Grier and Han22 also foundthat colloid-colloid interaction is related to the distancebetween confining walls. Further melting transition is also

a�Author to whom correspondence should be addressed. Electronic mail:[email protected]

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related to the colloid-colloid interaction.23 The glass slidesalso cause some boundary effect, for example, wall drageffect.20 The experimental setup of the system is complicatedand hard to assemble.20,21

Colloidal particles usually stay at an interface in a formof monolayer. Aveyard et al.2,3 reported some propertiesabout particle monolayer which was made of 2.6 �mcharged polystyrene particles at oil/water interfaces. Theseauthors also observed a solid-solid transition at relativelyhigh density induced by lateral compression. In comparisonwith the quasi-two-dimensional system, there are several ad-vantages in observing colloid particles in two dimensions bytrapping colloidal particles at oil/water interfaces. First of all,particles are trapped at the same plane and the vertical move-ment can be “almost” eliminated that ensures a two-dimensional system. It should be noted that the capillarywave fluctuations would permit the out-of-plane motion of acolloidal particle at oil/water interfaces. Since our oil/watersystem is far from its critical point, the amplitude of thecapillary wave fluctuations is relatively small compared tothe size of colloidal particles �2 �m in diameter�. Second,particles suspend at oil/water interfaces which can avoidboundary effects resulting from the walls in the quasi-two-dimensional system.

Recently, we applied Langmuir trough as a platform tostudy the phase behavior of a charged polystyrene colloidalparticle system in two dimensions at oil/water interfaces.24 Asequential phase transition, liquid phase→hexatic phase→solid phase, along with an increase in density, is observed.The purpose of this study is to explore the temperature effecton the phase behavior in the two-dimensional colloidal par-ticle monolayers. Digital video microscopy method is used tocapture the images of the configuration of particles. The pairdistribution function g�r� and bond orientation correlationfunction g6�r� of the system are evaluated by directly mea-suring particle locations in each frame of images. The phasebehavior of the system can be further determined by calcu-lating its structure factor S�k�. Further defect analyze is ac-complished by the Voronoi diagram to analysis the percent-age of different kinds of coordinated particles. It is foundthat the pair interaction potential of the system is a very longrange repulsion.

EXPERIMENTAL DETAILS

Decane �99% purity� was purchased from Merck andused as received. Water was purified by double distillationand then followed by a purelab Maxima Series �ELGA,LabWater� purification system with the resistivity alwaysbetter than 18.2 M� cm. The monodisperse polystyrene �PS�particles grafted with sulfate group with diameter of 2.000+0.052 �m were supplied by Interfacial Dynamic Corp.�USA� as a surfactant-free aqueous dispersion �solid content:8.1 g/100 ml�. The surface charge density on the particleswas 7.1 �C/cm2. Isopropanol �IPA� �99% purity� was pur-chased from Merck and used as a spreading solvent. The

spreading solution was prepared by mixing aqueous disper-sions of particles �as supplied� with IPA at the volume ratioof 1:3 or 1:6.

The experimental setup is schematically illustrated inFig. 1. The system of an oil/water interface was prepared ina glass vessel which was kept in a double-walled Pyrex ves-sel thermostated at a prescribed temperature by circulatingwater. The monolayer of particles at the oil/water interfacewas formed by injecting 0.5–6 �l spreading solution accord-ing to a prescribed surface density. The vessel was coveredby a glass slide for better temperature stability and to preventfrom contamination. An upward microscope �Olympus,BXFM� equipped with long working distance objectives withdifferent magnifications �5�, 10�, 20�, and 50�� was setup on the vessel for in situ observation. The observation wasperformed in dark field method.

Systems of different surface densities could be obtainedby carefully controlling the volume of the spreading solutioninjected onto the oil/water interface. The reduced surfacedensity �* was measured in units of particle number perframe area normalized by the cross-section area of particle,

�* = N��2

4A, �1�

where N is number of particles in each frame, � is the par-ticle diameter, and A is the surface area of each frame.

A long time observation was performed to examine theequilibration process of the system. After the spreading so-lution was injected into the interface, more than 50 frames ofimages every 1 h were recorded. There was no detectablechange of structure factor after 1 h. That implies that thesystem would reach equilibrium in 1 h. Therefore, all thedata in this study were collected after the system was left toequilibrate for at least 1 h.

The images were captured by a monochrome chargedcoupled device video camera �Sony Corporation� mountedon the microscope and grabbed by a frame grabber �NI-1409,National Instrument�. All these images were recorded by apersonal computer for further data analysis.

A standard image process was applied to extract the in-formation about positions of particles. We closely followed

FIG. 1. Schematic illustration of the experimental setup. An upward micro-scope is placed right above the vessel for in situ observation.

034706-2 B.-J. Lin and L.-J. Chen J. Chem. Phys. 126, 034706 �2007�


the method used by Crocker and Grier.25 After the positionsof particles are determined, the radial distribution functiong�r� is then calculated by

g�r� = �−2��i

�j�i

��ri��r j − r�� , �2�

where ��r� is the delta function. The bond orientational cor-relation function is defined and evaluated by

g6�r� = ��i

�j�i

�6i�ri��6j�r j − r�� , �3�

where

�6i =1

ni�j�i

ni

ei6ij , �4�

where ni is the number of nearest neighbors of particle i andij is the angle between two nearest neighbor bonds. Thelocal bond correlation function measures the deviation of thestructure near particle i from a perfect triangular lattice, forwhich ij =� /3 and �6i=1.

Once the radial distribution function is known, the struc-ture factor S�k� can then be calculated by the Fourier trans-formation of the total correlation function, h�r�=g�r�−1.

S�k� = � e−ik·rh�r�dr , �5�

where r is the position vector in the real space and k is theposition vector in the reciprocal space.

Voronoi polyhedra26,27 is applied to analyze the defectstructure in the two-dimensional particle monolayers. Notethat there is no defect species besides k=4, 5, 7, and 8 in ourexperimental results. The fraction of k-fold-coordinated par-ticles, Pk �k=4, 5, 6, 7, and 8� is thus defined as

Pk =nk

n4 + n5 + n6 + n7 + n8, k = 4,5,6,7, and 8, �6�

where nk is the number of k-fold-coordinated particles.In this study, digital video microscopy is also applied to

directly probe colloidal interactions.27 When the radial distri-bution function is determined in the limit of infinite dilution,the pair interaction potential can be evaluated through theBoltzmann distribution

U�r� = − kBT ln g�r� . �7�

On the other hand, the radial distribution function underthe condition of finite concentration also reflects neighboringparticles’ influence, and the potential of mean forces with itsstructure can be evaluated via

w�r� = − kBT ln g�r� . �8�

Note that the potential of mean force can be identified withthe system’s underlying pair potential only in the limit ofinfinite dilution,

U�r� = lim�→0

w�r� , �9�

where � is the density of the system. Reliable approxima-tions for U�r� can be obtained from w�r� using the Ornstein-

Zernike integral equation28 with appropriate closure rela-tions. In this study, the hypernetted chain approximation,29

U�r� = kBT�g�r� − 1 − c�r� − ln g�r�� , �10�

is applied to delineate the many-body effect to obtain thepair interaction potential U�r�. The function c�r� is thedirect correlation function defined by the Ornstein-Zernikeequation.

RESULTS AND DISCUSSION

Optical micrographs of PS particle monolayers at theoil/water interface are shown in Fig. 2 at four different re-duced surface densities, 0.004, 0.0155, 0.0257, and 0.0385,that correspond to four different states, a pure liquid phase, aliquid-hexatic-coexisting phase, a pure hexatic phase, and apure solid phase, respectively, observed in this study. Basedon the image analysis, the structure function, the angulardependence of the line shape of the two-dimensional struc-ture function, the radial distribution function, and the bondorientational correlation function are calculated and appliedto verify the phase behavior as a function of the surfacedensity. The calculated results are also illustrated in Fig. 2for four different states.

�1� The pure solid phase, �*=0.0385. The diffraction pat-tern shown in Fig. 2�d� demonstrates a clearly devel-oped sixfold angular pattern. The transverse line shapeof the first peak in the static structure function is welldescribed by a simple Lorentzian function, S�0�= ��0−�2+�2�−1, as shown in Fig. 2�d�, indicating thatthe system falls in the solid phase. The symbol 0 is theangular position of the first peak in the static structurefunction, is the in-plane angle that ranges from zeroto 2�, and � is the angular width of the simple Lorent-zian function. The translational order g�r� is quasi-long-range and decays algebraically. The bond orienta-tional order g6�r� is long range.

�2� The pure hexatic phase, �*=0.0257. The sixfold angu-lar symmetry illustrated in Fig. 2�c� is a necessary butnot sufficient condition to identify the phase behaviorof the system to be hexatic. The line shape of the firstpeak of the structure function in Fig. 2�c� fits very wellto a square-root Lorentzian function, S�0�= ��0−�2

+�2�−1/2. Note that the square-root Lorentzian func-tional form of the line shape of the first peak of thestructure function is well understood as a signature ofhexatic order.30 The translational order g�r� is shortrange and decays exponentially. The bond orientationalorder g6�r� is quasi-long-range and decays algebra-ically, in accord with the prediction of the KTHNYtheory.

�3� The liquid-hexatic-coexisting phase, �*=0.0155. Figure2�b� demonstrates a liquid-hexatic-coexisting phase,which is identified by the superposition of isotropicbackground intensity �for the liquid phase� and a

034706-3 Phase transitions in colloids at oil/water interface J. Chem. Phys. 126, 034706 �2007�


square-root Lorentzian function �for the hexatic phase�.�4� The pure liquid phase, �*=0.004. The diffraction pat-

tern shows in Fig. 2�a� a pattern of ring. The angulardependence of the diffraction peak is almost constan-

t.Since there is no angular dependence, this sample isan isotropic liquid. Both the translational order g�r� andthe bond orientational order g6�r� are short range anddecay exponentially.

FIG. 2. Experimental results for four different densities at 30 °C. Column �A�: �*=0.004, pure liquid phase; column �B�: �*=0.0155, liquid-hexatic-coexistingphase; column �C�: �*=0.0257, pure hexatic phase; and column �D�: �*=0.0385, pure solid phase. The first, second, third, fourth, and fifth rows of plots showthe optical micrograph, diffraction pattern, angular dependence of the line shape of the first peak, radial distribution function, and bond orientationalcorrelation function, respectively. In column �A�, the envelope of g�r� decays exponentially ��exp�−0.168r��, as well as the envelope of g6�r� ��exp�−0.504r��. In column �B�, the solid line of the angular dependence of the line shape of the first peak is fitted to the square-root Lorentzian function with abackground intensity, 9.91+4568�7.2912+ �−31.65�2�−0.5. The envelope of g�r� decays exponentially ��e−0.22r� and the envelope of g6�r� decays algebra-ically ��r−0.61�. In column �C�, the solid line of the angular dependence of the line shape of the first peak is fitted to square-root Lorentzian function,1260�2.0332+ �−30.50�2�−0.5. The envelope of g�r� decays exponentially ��e−0.19r� and the envelope of g6�r� decays algebraically ��r−0.27�. In column �D�,the solid line of the angular dependence of the line shape of the first peak is fitted to simple Lorentzian function, 2108�2.3422+ �−29.97�2�−1. The envelopeof g6�r� decays algebraically ��r−1.22�, while g6�r� is long range.



It should be pointed out that a sequence of phase transi-tions, pure liquid phase→ liquid-hexatic-coexisting phase→pure hexatic phase→pure solid phase, is observed alongwith an increase in density, in accord with the prediction ofthe KTHNY theory.12–16

The Voronoi polyhedron analysis is performed to deter-mine the fourfold-, fivefold-, sixfold-, sevenfold-, andeightfold-coordinated particles for each captured frame. Adefect site is defined as its coordination number other than 6.By the Voronoi polyhedron analysis, coordination number ofevery particle can be obtained to further analyze the defectpercentage.

Figure 3 shows the sample images at three different den-sities: �1� �*=0.0029, the liquid phase �Fig. 3�a��; �2� �*

=0.0137, the liquid-hexatic-coexisting phase �Fig. 3�b��; and�3� �*=0.0407, the solid phase �Fig. 3�c��, and their corre-sponding Voronoi diagrams. For clarity, in Fig. 3, all thesixfold-coordinated particles are marked by a blue dot. Onecan see in Fig. 3 that the fraction of sixfold-coordinated par-ticles dramatically increases from the liquid phase to thesolid phase.

One can see in Fig. 3�c� that there exist some defect sitesin the solid phase. Note that these defect sites are limited tofivefold- and sevenfold-coordinated particles. In addition,any fivefold-coordinated particle is always accompanied by asevenfold-coordinated particle, which is known as disloca-tion. The fractions of fivefold- and sevenfold-coordinatedparticles, P5 and P7, are also always smaller than 1.5% for

the system in the solid phase. Voronoi construction for thesystem in the hexatic phase is similar to that in the solidphase, except that more dislocations appeared in the hexaticphase. There are almost no fourfold- and eightfold-coordinated particles appearing in the hexatic phase.

Figure 4 shows the fractions of k-fold-coordinated par-ticles in Voronoi diagrams as a function of the reduced den-sity at three different temperatures: 20, 30, and 40 °C. It is

FIG. 3. Optical micrographs of PS particle monolayers at three differentstates and corresponding Voronoi diagrams at 20 °C: �A� pure liquid phase,�*=0.0013; �B� liquid-hexatic-coexisting, �*=0.0137; and �C� pure solidphase, �*=0.0407.

FIG. 4. Variation of the fraction of the k-fold-coordinated particles, Pk, as afunction of density �k=4, 5, 6, 7, and 8� at three different temperatures: �A�20 °C, �B� 30 °C, and �C� 40 °C. P4 ��, P5 ��, P6 ��, P7 ��, and P8

��.



obvious that the fraction of sixfold-coordinated particles in-creases along with the reduced density. Consider the systemat 20 °C, as shown in Fig. 4�a�. When �*=0.0013, the sys-tem falls in the liquid phase and P6=0.397. That means al-most 60% of particles are in defect structure. When the re-duced density is increased to 0.0052, the system goes intothe liquid-hexatic-coexisting region and P6=0.580. The sys-tem reaches to the pure hexatic phase at �*=0.019. When thereduced density is further increased up to the region of thesolid phase, the fraction of sixfold-coordinated particles P6

levels off and asymptotically becomes constant.The phase behavior at 20 °C, as shown in Fig. 4�a�, can

be classified into four regions: �1� �*�0.0045, the liquidphase; �2� 0.0045��*�0.0175, the liquid-hexatic-coexisting phase; �3� 0.0175��*�0.0208, the hexaticphase; and �4� 0.0208��*, the solid phase. It is obvious thatthe transition between the liquid and hexatic phases is firstorder due to the existence of a wide region of the liquid-hexatic-coexisting phase. On the other hand, the hexatic-solid-coexisting phase behavior is never observed in our sys-tem. Note that the density range of the hexatic phase is rathernarrow. It is very likely that the transition between the solidand hexatic phases is second order. It should be noted thatMarcus and Rice19,20 observed first order transition in bothliquid-to-hexatic and hexatic-to-solid phase transitions in thequasi-two-dimensional system of sterically stabilized un-charged polymethylmethacrylate spheres. This finding isconsistent with the prediction of the KTHNY theory.12–16 Inaddition, Bladon and Frenkel32 have reported the results ofMonte Carlo simulations of a two-dimensional system ofparticles with a pairwise additive potential consisting of ahard core repulsion and a very narrow square well attraction.These authors predicted that the liquid-to-hexatic phase tran-sition is first order while the hexatic-to-solid phase transitionmay be either first or second order.

Consider the transition from solid to hexatic phase basedon the defect analysis shown in Fig. 4. While the fraction offourfold- and eightfold-coordinated defects remain constantand almost zero, the fraction of fivefold- and sevenfold-coordinated defects start to increase right at the transitionfrom solid to hexatic phase. This implies that the transitionfrom solid to hexatic phase is induced by the substantialgeneration of fivefold- and sevenfold-coordinated defects. P5

and P7 keep increasing along with the further decrease indensity, even when the system falls into the liquid-hexatic-coexisting region. Note that P5 is always equal to P7 in theregions of liquid-hexatic-coexisting, hexatic, and solidphases, since one fivefold-coordinated particle and onesevenfold-coordinated particle always appear together toform a dislocation. Once the system density is further de-creased to enter the region of the pure liquid phase, P5 be-comes different from P7, in accord with results of molecularsimulation for quasi-two-dimensional system.31 In addition,the fractions of fourfold- and eightfold-coordinated defectsstart to increase substantially when the system enters thepure liquid region.

In addition to the system at 20 °C, we have carried outthe experiments at another two temperatures, 30 and 40 °C,and the results are also shown in Figs. 4�b� and 4�c�. Similar

to the results at 20 °C, the phase behavior at 30 and 40 °Ccan also be classified into four regions as a function of den-sity, as shown in Fig. 4. The phase boundaries for the systemat three different temperatures are plotted in Fig. 5. Note thatevery phase boundary shifts to a higher density as the tem-perature increases. The density window of the hexatic phaseremains quite narrow even at higher temperatures. On theother hand, the density window of the liquid-hexatic-coexisting phase enlarges along with rising temperature.

Consider a system of the solid phase at 20 °C, say, �*

=0.025. The melting transition can be observed by simplyincreasing the temperature. Once the temperature is raised to30 °C, the system would exhibit a transition from the solidto hexatic phase according to Fig. 5. If the temperature isfurther increased to 40 °C, the liquid phase would appear tocoexist with the hexatic phase. However, it is hard to tell,based on the results shown in Fig. 5, whether the hexaticphase vanishes at high temperatures. It is still expected thehexatic phase would vanish at certain higher temperaturesand the system would fall into the liquid phase due to thethermal fluctuations. To summarize the temperature effect onthe two-dimensional solid melting, the system under the con-dition of a fixed density would demonstrate a sequentialphase transition, pure solid phase→pure hexatic phase→ liquid-hexatic-coexisting phase→pure liquid phase, bysimply increasing the temperature. Similar phenomenon ofthe sequential phase transition has been also observed intwo-dimensional paramagnetic colloidal particles by varyingtemperature.33

It is interesting to point out that the solid phase occurs atextremely low densities �* 0.0208 in the system of 2 �msulfate-polystyrene particles at 20 °C. In contrast to the re-sults for the quasi-two-dimensional system of Rice andco-workers,19,20 the solid phase occurs at rather high densi-ties, �* 0.874 for PMMA particles and �* 0.704 for silicaparticles. That is because of the existence of very long-rangerepulsive forces between particles at the oil/water interface.These strong repulsions between particles at the oil/waterinterface lead to highly ordered structure even at relativelylow surface densities. This long-range colloid-colloid inter-action at oil/water interfaces has been studied by using anoptical tweezers method.10 In this work, we would like to

FIG. 5. The phase diagram �temperature vs density� for the two-dimensionalpolystyrene particles.



extract the pair interaction potential directly from the radialdistribution function via the Boltzmann equation, Eq. �7�.

As mentioned above, the pair interaction potential can beevaluated through the Boltzmann equation, Eq. �7�, in thelimit of infinite dilution. Therefore, we performed the experi-ments at some low surface densities. The radial distributionfunction for the system of �*=0.000 13 is shown in Fig. 6�a�,and this result is inserted into Eq. �7� to calculate the pairinteraction potential, as shown in Fig. 6�b�. In addition, thepair interaction potential U�r� /kBT calculated from Eq. �7�for the system of �*=0.000 21 is also illustrated in Fig. 6�b�.Note that the plots of the U�r� /kBT versus distance resultingfrom two different densities coincide with each other per-fectly. In addition, the hypernetted chain approximation, Eq.�10�, is applied to eliminate the many-body effect at lowdensities and this correction is negligible, as illustrated inFig. 6�b�. It is thus believed that the surface density of thesystem ��*=0.000 21� is low enough to be used to determinethe pair interaction potential. It is interesting to note that thepair interaction potential decays with distance to the powerof −3, as shown by the solid curve in Fig. 6�b�, consistentwith the finding of Aveyard et al.10 These authors applied an

optical tweezers method to find out that the force betweenparticles decays with distance to the power of −4, that im-plies that the pair interaction potential decays with distanceto the power of −3. This long-range repulsion arises prima-rily from the presence of a small residual electric charge atthe particle-oil interface.10

In addition to the system at 20 °C, we have carried outthe experiments to calculate the pair interaction potential un-der the condition of low surface density ��*=0.000 13� atother two temperatures, 30 and 40 °C. The pair interactionpotentials, U�r� /kB, as a function of distance r for three dif-ferent temperatures are shown in Fig. 7. The results of threedifferent temperatures perfectly coincide with one another, asexpected, to form a single curve. In addition, the hypernettedchain approximation, Eq. �10�, is applied to eliminate themany-body effect at three different temperatures and this cor-rection is negligible, as illustrated in Fig. 7. Note that thispair interaction potential also decays with distance to thepower of −3.

CONCLUSIONS

The two-dimensional charged sulfate-polystyrene par-ticle �2 �m in diameter� monolayer at decane/water inter-faces is chosen to study the equilibrium structure by usingan enhanced digital video microscopy. A sequentialphase transition, pure solid phase→pure hexatic phase→ liquid-hexatic-coexisting phase→pure liquid phase, is ob-served as decreasing surface density at three different tem-peratures: 20, 30, and 40 °C. The density window of theliquid-hexatic-coexisting region enlarges as temperature israised. The same sequential phase transition is also observedfor the system of a fixed density by simply raising the tem-perature. In addition, the liquid-hexatic phase transition isfirst order, while the solid-hexatic phase transition is secondorder.

The Voronoi construction is applied to analyze the defectstructure in the two-dimensional particle monolayers. Thetransition from solid to hexatic phase is induced by the sub-

FIG. 7. The pair interaction potential function resulting from the Boltzmannequation, Eq. �7�, at 20 °C ��, 30 °C ��, and 40 °C ��. Blue solid curveis fitted to the equation U�r� /kB=93 668�d /r�3 and red solid curve repre-sents the U�r� /kB evaluated from the HNC approximation.

FIG. 6. �A� The radial distribution function for the system of �*=0.000 13 at20 °C. �B� The pair interaction potential function resulting from the Boltz-mann equation, Eq. �7�, at two different surface densities: �*=0.000 13 ��and �*=0.000 21 ��. Blue solid curve is fitted to the equation U�r� /kBT=294.9�d /r�3 and red solid curve represents the U�r� /kBT evaluated fromthe HNC approximation at �*=0.000 13.



stantial generation of fivefold- and sevenfold-coordinated de-fects. When the system density is decreased to enter the pureliquid region, the fraction of five-coordinated defects P5 be-comes different from that of sevenfold-coordinated defectsP7 and the fractions of fourfold- and eightfold-coordinateddefects start to increase substantially. The pair interactionpotential of the two-dimensional colloidal particles is foundto be a very long range repulsion and to decay with distanceto the power of −3, consistent with the result from the opticaltweezers measurement.10

ACKNOWLEDGMENT

This work was supported by the National Science Coun-cil of Taiwan.

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