Effect of Molecular Weight on the Capillary ... - ACS Publications

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Effect of Molecular Weight on the Capillary Absorption of Polymer Droplets Srikanth Dhondi,*,† G. G. Pereira,‡ and Shaun C. Hendy†,§ †

School of Chemical and Physical Sciences, MacDiarmid Institute for Advanced Materials and Nanotechnology, Victoria University of Wellington, Wellington 6011, New Zealand ‡ CSIRO Mathematics, Informatics & Statistics, Private Bag 33, Clayton South 3169, Australia § Industrial Research Ltd., Lower Hutt, New Zealand ABSTRACT: We study capillary absorption of small polymer droplets into nonwettable capillaries using coarse-grained molecular dynamics simulations and a simple analytical model. Studies of droplets of simple fluids have revealed that the capillary process depends on the ratio of tube-to-droplet radii [Willmottet al. Faraday Discuss., 2010, 146, 233; Marmur J. Colloid Interface Sci. 1988, 122, 209]. Here we consider the absorption of droplets of polymers and study the effect of polymer chain length on the capillary absorption process. Our simulations reveal that for droplets of the same size (radius), the critical tube radius, below which there is no absorption, increases with the length of the polymer chains that constitute the droplets. We propose a model to explain this effect, which incorporates an entropic penalty for polymer confinement and find that this model agrees quantitatively with the simulations. We also find that the absorption dynamics is sensitive to the polymer chain length. In some cases during the capillary uptake transient partial absorption states, where the droplet is partially in and partially out of the tube, were observed. Such dynamics cannot be explained by a generalized Lucas−Washburn approach.

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scale protein patterns,17 nanofibers,18 nanobelts,19 etc. are produced using capillary forces. In most of these experiments we encounter situations that fall in the above-described regime: that is, the size of the liquid reservoir is comparable to the channel/tube dimensions. Thus, in these situations, the Laplace pressure can become important in influencing the capillary phenomena. The behavior of polymer solutions or melts at the nanoscale can be particularly more complicated because at these dimensions the characteristic length scales of polymers become comparable to the device dimensions. When a single polymer molecule is forced to enter a tube whose diameter is comparable to its size then the molecule has to stretch in the direction of the tube which will result in a conformational entropy loss. This can affect the behavior of polymers in these devices and can lead to new phenomena.20,21 Therefore, it is legitimate to ask the question, “What is the effect of molecular weight on the capillary absorption of polymer droplets?” In the context of nanocapillaries, a question that received attention, recently, is whether the Lucas−Washburn equation3,4 for capillary rise applies at the nanoscale. This is yet to be fully resolved, with the literature containing conflicting results.2,22−28 In addition, the entropic penalty associated with the capillary absorption of polymer droplets may cause the capillary

INTRODUCTION Capillary forces have a wide range of applications in nature and industry.1,2 While the Lucas−Washburn model3,4 has been found to provide an effective description of capillary absorption from a large reservoir, the dynamics are altered if a capillary is brought into contact with a droplet.1,5,6 Indeed in many micro/ nanofluidic systems, the size of the liquid reservoir is often comparable to the device dimensions. In such situations, to describe the system behavior correctly finite size effects must be taken into account. Marmur1 was the first to study the capillary absorption of droplets of simple liquids. He argued that the Laplace pressure originating from the curvature of the droplets could lead to the capillary absorption of nonwetting liquids if the droplets are sufficiently small.7−9 This is in contrast with the standard macroscopic theories,3,4,10,11 according to which there should be no capillary action for nonwetting liquids (contact angles greater than 90°). Later, Schebarchov and Hendy6 were able to derive and solve a simple generalization of the Lucas− Washburn equation to describe the dynamics of the droplet uptake problem. As yet, these results have yet to be extended to complex fluids such as polymer solutions or melts. Indeed, there is a growing interest in building polymer-based functional nanomaterials and nanostructures for novel applications.12−14 One of the methods used for producing these nanostructures is by filling nanomolds with the aid of capillary forces. For example, polymer-based nanowires,15 nanorods and nanotubes,16 nano© 2012 American Chemical Society

Received: March 1, 2012 Revised: May 28, 2012 Published: June 1, 2012 10256

dx.doi.org/10.1021/la300903w | Langmuir 2012, 28, 10256−10265

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dynamics to be different from simple fluids. Furthermore, entanglement effects may play an important role in influencing the underlying process. Melts consisting of shorter chains exhibit Rouse-type dynamics whereas those with longer chains showcase reptation-type dynamics.29,30 Therefore, it is of considerable interest to study the effect of macromolecular nature of polymers on the thermodynamics and absorption dynamics of the capillary absorption process. For this purpose, we carry out molecular dynamics (MD) simulations of nonwetting polymer droplets. First, we study the effect of molecular weight, for two different droplet sizes, on the contact angle for a nonwetting case by means of contact angle simulations. Next we consider capillary absorption simulations of droplets of the same size but made up of chains of different lengths (N) and study the impact of molecular weight on the capillary process. We restrict our results to the two cases N = 20 and 200 as these chain lengths are far apart enough to demonstrate considerable differences in the observed capillary processes. We then present a simple analytical model to interpret the simulation results.

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Generating Equilibrium Droplet Configurations. The initial droplet configurations were generated by placing constituent monodisperse chains randomly in a given spherical volume, to realize the system at approximately 0.85 density. The initial bond length (b) between the monomers was set to 0.97σ. Because of random placement of the chains, the initial configurations consist of overlapping monomers. From this state, the system was driven toward equilibrium using the soft potential34 ⎧ for r ≤ rcs ⎪ A[1 + cos(πr / rcs)] Φsoft (r ) = ⎨ ⎪ for r > rcs ⎩0

where r is the distance between any two monomers (bonded or nonbonded monomers) and rcs = 21/6σ is the cutoff radius for interaction. This potential ensures that monomers that are too close (r → 0) and that are separated by nearly a bond length (r → b ≈ rcs) do not experience large forces while relatively large forces are exerted on monomers separated by intermediate distances between 0 and b. The prefactor A in (eq 3) is the amplitude of the interaction energy whose value was steadily increased in time in such a way that the overlapping monomers are pushed apart in a gradual fashion. The time integration of the equations of motion was carried out using the velocity Verlet algorithm.35 Note, during this equilibration period there was no LJ interaction between monomers. The temperature of the system was controlled at T = 1.0ε/kB using the Langevin thermostat36,37 with the corresponding equation of motion for any particle i is given by

METHODOLOGY

In this section, we discuss the interaction potentials, the steps involved in preparing the equilibrium droplet configurations and substrate/tube details. All the simulations reported in this work are molecular dynamics simulations carried out using the classical molecular dynamics package LAMMPS.31 Potentials. The polymer molecules are modeled as coarse grained bead−spring chains where each chain consists of Neffective monomers. The interaction between bonded monomers is given by the finitely extensible nonlinear elastic (FENE) potential:30,32 ⎧ ⎡ ⎛ r ⎞2 ⎤ 1 2 ⎢ ⎪ ⎪− kr0 ln 1 − ⎜ ⎟ ⎥ for r ≤ r0 ⎢⎣ ⎝ r0 ⎠ ⎥⎦ ΦFENE(r ) = ⎨ 2 ⎪ ⎪∞ for r > r0 ⎩

mrï + m Γri̇ = −∑ Fij + fi i≠j

(4)

where Γ = 1.0τ−1 is the friction constant, f i is the random force applied on monomer, i and Fij = −∇Φij is the total force experienced by particle i due to its interaction with particle j. Here τ = (mσ2/ε)1/2 is the LJ characteristic time scale. Using the soft potential the system was initially relaxed for about 6τ to 8τ, with a time step Δt = 10−4τ. After this relaxation period the energy of the droplets attained approximately constant value, suggesting that the system has obtained equilibrium. At this point the soft potential between the monomers was replaced by the LJ potential (eq 2). The system was further equilibrated for 105τ with Δt = 0.01τ. Substrate/Tube. The substrates used in the contact angle simulations are single atom thick, simple cubic layers with a lattice constant 1.0σ. The capillary tubes used in the absorption simulations were constructed by rolling a single atom thick, simple cubic layer, with lattice constant 1.0σ, into tubes with both ends open. To reduce the computational cost, substrate or tube atoms were fixed in space throughout all simulations presented in this article. Although, in reality, the substrate or the tube atoms vibrate about their equilibrium positions this should not influence the nature of the results since all our simulations were conducted under the same condition (stationary substrate or tube). Moreover, simulations of confined liquids with rigid or mobile tube walls were shown to give similar results.38,39

(1)

Here the parameters r0 = 1.5σ and k = 30εσ−2 are the maximum bond extension and the spring constant respectively; σ and ε are the length- and energy-scales in the Lennard-Jones units. The above choice of parameters ensures that there is no bond breaking or chain crossing.33 The short-range repulsion between monomer−monomer (both bonded and nonbonded) and monomer−substrate/tube atom pairs is implemented via a shifted Lennard-Jones (LJ) potential: ⎧ ⎡ 12 12 6 ⎪ 4ε⎢⎛⎜ σ ⎞⎟ − ⎛⎜ σ ⎞⎟ − ⎛⎜ σ ⎞⎟ for r ≤ r c ⎪ ⎢⎝ r ⎠ ⎝r⎠ ⎝ rc ⎠ ⎣ ⎪ ⎪ ⎛ σ ⎞6 ⎤ ΦLJ (r ) = ⎨ ⎪ +⎜ ⎟⎥ ⎝ rc ⎠ ⎥⎦ ⎪ ⎪ ⎪0 for r > rc ⎩

(3)

(2)

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where r is the distance between monomers and rc = 2.5σ is the cutoff radius for interaction. In all our simulations we have used mass m = 1.0, ε = 1.0, and σ = 1.0. All the physical quantities used in this article are expressed in LJ units, unless otherwise specified.

CHARACTERIZATION OF THE DROPLET The equilibrium properties of the droplets, that are of interest to our study-, were extracted by recording the droplet configurations every 100τ for 5 × 104τ. First of all we calculate 10257


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CONTACT ANGLE MEASUREMENTS The contact angle, θ, between the liquid and solid surfaces can be used to discriminate between wetting and nonwetting surfaces. Within the framework of MD simulations control over the contact angle can be gained by tuning the interaction between the liquid and solid constituents. By varying the strength of this interaction, the wetting properties of the liquid with respect to the solid can be changed. Equilibrated polymer droplets, obtained using the procedure described in the previous section, are placed on a solid substrate. The interaction between the substrate atoms and monomers was described by the shifted LJ potential (eq 2). Here we are only interested in nonwetting contact angles, for the following reasons. It has been shown that in wetting situations, Newtonian fluids will always be absorbed into tube irrespective of the tube radius and there exists a critical tube to droplet radius only in nonwetting cases.8,9 Moreover, we found that polymer chains crawl on the outer surface of the tubes in wetting cases. For these reasons we restrict ourselves to nonwetting cases and as we will show, an LJ interaction strength of εps = 0.5ε fulfills this condition (θc > 90°) in our simulations. So we fix the interaction between the polymer− substrate/tube (εps or εts) at 0.5ε for the reminder of this article. The temperature of the droplets was maintained constant at 1.0ε/kB (kB, Boltzmann’s constant), by coupling the monomers to the Langevin thermostat, with a friction coefficient Γ = 1.0τ−1. Although the application of the Langevin thermostat alters the dynamics of the system, this does not cause serious concern in this particular case, as we are only interested in the equilibrium contact angles. Contact angles were extracted by approximating the equilibrated droplet on the substrate to a spherical cap. If a droplet with an initial radius r0 assumes a spherical cap geometry of height h upon reaching equilibrium then the contact angle formed by the liquid at the substrate surface is given by

the average radius of the droplets with the same number of total monomers but made up of chains of different length. (For example, for a droplet of 8000 monomers we ran separate simulations of 400 chains of length 20, 80 chains of length of 100, and so on.) The average droplet radius ⟨r0⟩ was 15.03 ± 0.07σ for droplets with 8000 monomers and 18.58 ± 0.07σ for droplets with 16000 monomers (where the error represents the standard deviation computed over all considered chain lengths). The narrow distribution in ⟨r0⟩ indicates that there is very little effect of chain length on the average droplet size. This is useful to know since we want to study the capillary action of droplets of same size (same radius) but made up of chains of different lengths. For droplets of same size, the local environment experienced by an average chain should change with chain length, N. To investigate this effect we calculated the average radius of gyration of the chains in the droplet as a function of N and the results for 8000 monomer droplets are shown in Figure 1. In

Figure 1. Log−log plot of ⟨Rg⟩ as a function of chain length N for two different cases. The black curve (circles) represents ⟨Rg⟩ of a chain in a droplet consisting of 8000 monomers and the red curve (squares) is the data from single chain simulations.

⎞ ⎛ 3 ⎟ θc = cos−1⎜1 − 3 3 4r0 /h + 1 ⎠ ⎝

(5)

The droplets were initially equilibrated for 10 τ on the substrate to let them reach their equilibrium state (height). After this time period the droplet configurations were noted down every 100τ for the next 4 × 104τ to calculate the average height (⟨h⟩) of the droplet. The average radius of the droplet ⟨r0⟩ are available from the previous section. By substituting ⟨h⟩ and ⟨r0⟩ into (eq 5) approximate θc’s were extracted for each N case. The contact angle measurements for different chain lengths, for two different droplet sizes are plotted in Figure 2. The contact angles are roughly constant independent of the chain length or the droplet size, within the studied chain length range. This result is in qualitative agreement with the earlier observations.40,41 We estimate the average contact angle for the case of εps = 0.5ε as θc ≈ 104° within 1% error. 4

the small N limit on average the chains behave as if they were ideal chains, i.e., ⟨Rg⟩ ∝ N1/2. There appears to be a crossover from this behavior for N > 200. A possible explanation for this may be as follows. In small N droplets, on average each chain is surrounded by a large number of chains, approximately equally in all directions. This is analogous to a chain in a melt, which is ideal. As N increases on average each chain will have fewer polymer neighbors, i.e. more exposure to solvent. Hence the trend in ⟨Rg⟩ deviates from the ideal chain behavior as N increases. In order to gain better understanding of this result we carried out single chain simulations, which were performed under the same conditions (temperature, monomer−monomer interactions) as the droplet simulations above. The ⟨Rg⟩ data from these simulations is compared against the droplet simulations (represented by red line or squares in Figure 1). The single chain simulations reveal that the chains display poor solvent behavior, where ⟨Rg⟩ ∝ N1/3. In the limit N → system size (total number of monomers in the droplet), the ⟨Rg⟩ curve from the droplet simulations converges onto the single chain curve. This observation supports our earlier explanation for the shrinkage in ⟨Rg⟩ as N increases. Similar ⟨Rg⟩ behavior was observed for larger droplets of 16000 monomers.

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RESULTS For the capillary uptake simulations, the droplets were placed at the entrance of one of the tube ends. The tube direction is chosen to be along the y axis. The interaction between the tube atoms and monomers was described by the shifted LJ potential (eq 2) with parameters σpt = 1.0σ and εpt = 0.5ε. The Langevin thermostat is a good choice to study equilibrium properties such as radius of gyration and contact angle measurements 10258


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Figure 2. Approximate contact angles as a function of chain length for two different droplet sizes is shown here. The black curve (circles) represents the data from 8000 monomer droplets and the red curve (squares) represents the data from 16000 monomer droplets.

Figure 3. Top: the meniscus height (axis right). Bottom: radius of the protruding droplet (axis left). These are plotted as a function of time for two different radii of the tube, for an 8000 monomer droplet with N = 20 chains.

discussed in the previous sections. However, because of its stochastic nature, it is not preferred when it comes to studying dynamic properties. On the other hand the Nosé−Hoover thermostat gives better dynamics compared to Langevin thermostat42 and hence for this reason we switch to the Nosé−Hoover thermostat for all capillary absorption simulations. The temperature of the droplet was kept constant at T = 1.0kB/ε using the Nosé−Hoover thermostat36 with a damping constant 10τ. For a droplet of fixed size (8000 monomers or 16000 monomers), for both chain cases N = 20 and N = 200, we ran a series of simulations at different tube radii, rt. The tube radius below which there is no absorption was identified as the critical tube radius rtc. For a fixed droplet size, first we identify rtc for N = 20 case followed by N = 200 case. As we will show, since the interesting physics only takes place in the vicinity of rtc, we restrict our discussion to only those cases where rt is close to rtc.

⟨R gx 2⟩ = ⟨R gy 2⟩ = ⟨R gz 2⟩ =

N

1 N

∑ (xi − Xcm)2 ;

1 N

∑ (yi − Ycm)2 ;

1 N

∑ (zi − Zcm)2

i=1 N i=1 N i=1

(6)

where x, y, and z are the components of the monomer position vectors and Xcm, Ycm, and Zcm are the coordinates of the center of mass of the respective chain. Note that the tube direction is along the y axis. The average behavior of each of these components should capture any interplay between the molecular aspect of polymers and the absorption process. Figure 4 shows the ⟨Rg⟩ components as a function of time for both rt = 5.10σ and rt = 5.26σ. The average size of chains does not change in the case of rt = 5.10σ. In the case of rt = 5.26σ,

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DROPLETS WITH 8000 MONOMERS A number of simulations were run for different tube radii between rt = 4.26σ to 6.69σ and the subsequent capillary process for each of the cases was monitored by recording the height of the liquid meniscus and the radius of the protruding droplet as a function of time. The results for the N = 20 case are shown in Figure 3. For the smaller tube radius of rt = 5.10σ, both the radius of protruding droplet and the height of liquid meniscus remain approximately constant close to 13σ and 0σ respectively. But for a slightly larger radius of rt = 5.26σ the droplet gets absorbed into the tube. In this case it can be seen that the height of the meniscus increases and the radius of the protruding droplet decreases. In order to determine the critical tube radius, the simulation with smaller tube radius of rt = 5.10σ was run for much longer and yet no absorption was seen. This indicates that the critical tube radius (rtc) for 8000 monomer droplet with N = 20 chains is 5.26σ. To study the impact of absorption on the molecular aspect of the polymers we calculated the average size of the chains, averaged over all the chains in the droplet. This was accomplished by computing all the three components of the radius of gyration, Rg

Figure 4. Average values of the components of Rg are plotted as a function of time for tube radii rt = 5.10σ and rt = 5.26σ for the 8000 monomer droplet with N = 20 chains. 10259


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the Rgy component increases during absorption while the other two components decrease at the same time. This indicates that during the absorption the chains elongate in the tube direction while becoming compressed in the other two directions. This will result in a configurational entropy loss, which will lead to an entropic penalty for absorption, although it is not clear how significant a role this will play in the process. We have also investigated the effect of absorption on the potential energy, the sum of both bonded (FENE) and nonbonded (LJ) contributions. Figure 5 shows the potential

Figure 6. Meniscus height and the radius of the protruding droplet as a function of time for the 8000 monomer droplet with N = 200 chains, for different tube radii.

the macromolecular nature of the liquid. The droplet eventually overcomes this barrier and gets drawn up the tube. The average chain size behavior for N = 200 and rt = 5.57σ case is shown in Figure 7. Prior to absorption, while the droplet

Figure 5. Potential energy behavior for the 8000 monomer droplet with N = 20 chains for different tube radii.

energy for the above two cases. The potential energy remains approximately constant in the case of rt = 5.10σ, as you would expect when there is no absorption, whereas it increases in the case of rt = 5.26σ during the absorption. The sharp fall in potential energy in the case of rt = 5.26σ occurs right after total encapsulation of the droplet. This indicates that there is an energy barrier associated with absorption. Similar calculations were carried out for a series of N = 200 droplets to examine whether an increase in molecular size has any effect on the whole process. The meniscus height and the radius of the protruding droplet for this case, for different tube radii, are shown in Figure 6. Several important observations can be drawn from this plot. First of all the critical tube radius rtc has shifted to a higher value 5.57σ compared to 5.26σ in the case of N = 20. This indicates that the molecular weight of the polymers has an effect on the capillary absorption. In the case of rt = 5.57σ the droplet sits at the tube entrance for a time, during which it tries to enter the tube by overcoming some sort of barrier. This is clearly evident from the meniscus height plot (see Figure 6). The origin of this barrier could be energetic (as observed by Schebarchov and Hendy in Newtonian droplets8) or entropic resulting from the conformational entropy loss or a combination of both. Note that in this case the tube radius 5.57σ is comparable to the average chain size 5.83σ. Thus, considerable elongation of chains may take place during absorption resulting in a conformational entropy loss. During the early times transient partial absorption, where the droplet is partially in and partially out of the tube, was noticed. Such a process was not seen in the earlier N = 20 case and to the best of our knowledge has not been observed for either Newtonian or non-Newtonian fluids earlier. We attribute this effect to the increase in molecular size N or in general, to

Figure 7. ⟨Rg⟩ components for rt = 5.57σ case for 8000 monomer droplet with N = 200 chains. The blue dashed encircled regions show the partial absorption instances. The partial absorption and desorption process is demonstrated with the help of snapshots of the simulation.

attempts to enter the tube, fluctuations in Rgy were observed and this trend is in agreement with the partial absorption and desorption pattern seen in the meniscus height (refer Figure 6). Corresponding to continual absorption or desorption there was either an increase or decrease in ⟨Rgy⟩ and consequently a decrease or increase in ⟨Rgx⟩ and ⟨Rgz⟩, as demonstrated in Figure 7. Hence the average behavior of the Rg components gives a good description of the underlying process on the molecular aspect of polymers. The average chain size measurements were also performed for the wider tube rt = 5.73σ and the results are shown in Figure 8. In this particular case, we were able to study the system even after complete absorption, due to the faster dynamics compared to the rt = 5.57σ case. Once the droplet was totally encapsulated 10260


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description as in,6 which is a model based on the standard Lucas−Washburn equation for describing the capillary dynamics of finite droplets. According to this model, once the flow has reached steady state, the meniscus height of a droplet of a Newtonian fluid during absorption can be found using the following equation: cos θc ⎞ dh(t ) 8μ 2γ ⎛ 1 = + ⎜ ⎟ (rt + 4b)rt dt h(t ) ⎝ rd(t ) rt ⎠

(7)

Here γ is the surface tension, rd(t) is the radius of the protruding droplet at time t, b is the slip length, and μ is the viscosity of the fluid. To reduce the mathematical complexity, it is convenient to solve the corresponding equations in terms of dimensionless variables H ≡ h/rt, Rd ≡ rd/rt, and φ ≡ t/ts, where ts ≡ ((4μrt)/(γ(4b/rt + 1))) is the characteristic time scale of the flow. Using these definitions, the (eq 7) can be rewritten in dimensionless form as Figure 8. ⟨Rg⟩ components for rt = 5.73σ case for 8000 monomer droplet with N = 200 chains. The blue vertical dashed line separates the absorption process into before and after complete encapsulation.

H

dH 1 = + cos θc dφ Rd

(8)

Assuming that the volume of droplet is approximately conserved during the capillary uptake, we arrive at

into the tube, the meniscus height remained approximately constant (refer to Figure 6), during which period we monitored the average Rg components and the results are shown in Figure 8. During the absorption, similar to N = 20, the chains elongate in the tube direction while shrinking in the other two directions. After complete absorption the Rgy component starts to decrease with time whereas Rgx and Rgz components increase relatively slowly during the same period. In the long time limit, we expect the chains to attain their original size as in the bulk (unless obstructed by the capillary walls). The potential energy measurements complement this argument for the relaxation process (see Figure 9). Throughout the transient absorption

3H ≈ 4(R 0 3 − Rd 3)

(9)

Using eq 9, eq 8 can be rewritten just as a function of Rd(φ), whose analytical form is φ − φ0 = −

16 3

∫R

Rd 0

Rd3(R 0 3 − Rd 3) dR (1 + Rd cos θc)

(10)

where φ0 is the chosen initial time and R0 is the droplet radius at φ0. Thus obtained Rd can be plugged back into (eq 9) to retrieve the meniscus height H. Our simulation data, in general, do not fit very well with the model (eq 10). As an example in Figure 10 we show the best fit cases, a droplet made up of N = 20 chains for rt = 5.26σ case and a droplet made up of N = 200 chains for rt = 5.73σ. In both these cases the model is unable to fit 10% of the data, although it fares reasonably well during the very late stages of the absorption. One possible explanation for this poor performance

Figure 9. Potential energy for 8000 monomer droplet with N = 200 as a function of time, for different tube radii.

states, depending on whether the droplet is inside or outside the tube, the potential energy either increases or decreases, respectively. After the complete absorption the potential energy quickly falls and attains a new value. This sharp fall in potential energy reaffirms the existence of energetic barrier, also observed in the N = 20 and rt = 5.26σ case. Dynamics. To understand the capillary dynamics observed in our simulations, we follow the generalized Lucas−Washburn

Figure 10. The data points indicate the MD meniscus height data and the solid curves are the model fits to eq 10, for two different cases. The model at most can describe only very late stage dynamics thus the fits are restricted to this region of the MD data. 10261


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is that during the initial times there are significant entanglements in the system which are not included in the model. However, toward the end of the absorption the polymer chains take up extended configurations (as evident from the Rg data earlier) reducing the entanglement effects, and giving the reasonable fits by the model during this stage of dynamics. However, the model completely fails in situations like in the case of 8000 monomer droplet made up of N = 200 chains for the tube radius rt = 5.57σ. As shown, in this case we observe continual partial absorption and desorption during the initial times which cannot described by the generalized Lucas− Washburn approach. Droplets with 16000 Monomers. Absorption simulations were also performed on the larger droplets of 16000 monomers, for chain lengths 20 and 200. Once again the critical tube radius was identified by running a series of simulations over a broad range of tube radii (4.94−7.49σ). The height of the meniscus and the radius of the protruding droplet for N = 20 droplet are shown in Figure 11 and the

Figure 12. Meniscus height as a function of time for the 16000 monomer droplet with N = 200, for various tube radii. In the inset we highlight the metastable and partial absorption regions for the case of rt = 6.69σ.

several metastable states, where the droplet gets stuck for considerably long periods of time, were observed during absorption. Such a process contrasts with any of the existing models3,4,6 aimed at explaining capillary dynamics and certainly cannot be described by the generalized Lucas−Washburn equation (eq 10). The results from this section are summarized in the Table 1. Table 1. Critical Tube Radii Listed as a Function of Droplet Size and Chain Length droplet size (no. of monomers) 8000 16 000

Figure 11. Meniscus height and radius of the protruding droplet plots for the 16000 monomer droplet with N = 20 chains, for various tube radii.

N

⟨Rg⟩σ

Rtcσ

20 200 20 200

2.14 5.83 2.14 5.83

5.26 5.57 6.37 6.69

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THEORETICAL ANALYSIS The MD results from the previous sections show for droplets composed of longer chains, a larger critical tube radius is required than for droplets composed of shorter chains. Specifically, we have found that the radius of gyration of the N = 20 chains is 2.14σ while the N = 200 chain is 5.83σ, compared to critical tube radii of roughly 5.26σ and 5.57σ (respectively) for the droplet composed of 8000 monomers and 6.37σ and 6.69σ (respectively) for the droplet composed of 16000 monomers. We shall now attempt to understand the previous MD simulation results from a theoretical point of viewwhy do the droplets composed of longer chains have a larger critical tube radius? We only consider equilibrium thermodynamics here and therefore only model the initial and final, equilibrium conformations of the polymer droplets. Previously Schebarchov and Hendy8 were able to show that nonwetting metallic droplets could be drawn up a narrow capillary tube if the droplet radius was sufficiently small. Their theoretical analysis was based on surface tension arguments (between the metallic droplet, the tube walls and the surrounding solvent) which gives rise to a Laplace pressure which assists the droplet in rising up the tube. Our analysis is based on these ideas, however, the complicating factor is that

critical tube radius was identified as 6.37σ. The activation time, that is the time after which the absorption process starts, has increased considerably compared to the 8000 monomer droplet of same chain length. This could be due to the decrease in curvature of the droplets resulting in weaker Laplace forces or a much longer time for chains to disentangle. The meniscus height data for the N = 200 droplet is shown in Figure 12 and the critical tube radius is around 6.69σ. Once again for the same droplet size the critical tube radius increases, from 6.37σ for N = 20 to 6.69σ for N = 200. The capillary dynamics observed in this case differs substantially from the any of the earlier cases discussed. In the case of rt = 6.53σ throughout the simulation period continual partial absorption and desorption of the droplet was observed. It appears that the droplet is unable to overcome the barrier for absorption during the simulation period. One can argue that this is to do with the slow dynamics associated with longer polymers and given sufficiently long time the droplet may eventually overcome this barrier to enter the tube. Though we do not deny this possibility, the interesting outcome from this result is the persistent partial absorption and desorption of the droplet, which has not been reported before. In the case of rt = 6.69σ 10262


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Next, we consider the situation when the droplet is completely absorbed within the capillary tube. We make the following assumptions in modeling this state: • We assume the chains have sufficient time to disentangle and form separate blobs when moving into the tube. In doing this, they attempt to minimize the conformational entropy penalty. This, of course, may not always be kinetically possible but for sufficiently short chains (remember we only simulated absorption for chains up to N = 200) is reasonable. • The tube radius, rt, needs to be smaller than or at least comparable to the radius of gyration of the individual chains. If the tube radius is much larger than the radius of gyration of the individual chains, then there will be no entropy penalty for entering the tube. • We assume the chains within the droplet behave like ideal chains even though the whole droplet is in a poor solvent. Our MD simulations validate this assumption (for chains that are short enough). A schematic of a fully absorbed droplet is shown in Figure 13c. The volume of this droplet is

for polymer chains one needs to account for chain entropy. If a chain is confined within a narrow tube the number of possible conformations it may investigate decreases and as a result the chain’s free energy increases. Figure 1 shows that for N small the radius of gyration of the polymer chains which make up the droplet scales roughly like a random walk chain, i.e., an exponent of 0.484 compared to the random walk exponent of 0.5. Moreover, for almost all chain lengths sampled their radius of gyration is significantly larger than a single polymer chain (of the same length) in a poor solvent. This indicates the chains are much more extended and investigate many more configurations than a single polymer chain in a poor solvent. Even though the droplet is in a poor solvent the individual chains which make up the droplet behave like ideal chains. This is true as long as N is sufficiently small (N = 200 being an upper limit). We consider the situation where we have a droplet with a constant number of total monomers. Then, given that the droplet is composed of chains of N monomers each, we shall denote the number of chains in the droplet as m and we keep mN constant. We assume the chains which make up the polymer droplet are ideal (random-walk) chains and we measure chain entropy with respect to this state. Consider Figure 13a, which shows the situation when the droplets sit at the lower end of the capillary. The Helmholtz

⎛1 − 4 VII = πrt L − πrt 3⎜⎜ 3 ⎝ 2

3 2

sin θc + cos3 θc

1 3

sin 3 θc ⎞ ⎟ ⎟ ⎠

(13)

and of course this volume is just mNb3. Using this equation we can eliminate L in favor of rt. The surface energy associated with this droplet shape is FIIS = γSL 2πrtL + γLV

⎛ ⎡ ⎤2 ⎞ b Fconfined = kBT ⎜⎜N ⎢ ⎥ − 1⎟⎟ ⎝ ⎣ rt ⎦ ⎠

free energy of this state, denoted FI, is just the surface energy of the droplet, since we measure chain entropy with respect to the ideal state. Thus, we have

(15)

We should mention the N in the above equation does not directly correspond to the chain lengths we used in the MD section. The MD represents a coarse-grained model where several base units are grouped into each monomer. As a result the N in (eq 15) is on the order of 103−106 and so the last term of minus one is negligible compared to the first term. However, we will leave it here since it has some significance (see below). Recall we have m chains which make up this droplet so the total entropic energy penalty, FPII, is

(11) 3

and note that due to volume conservation mNb = 4πrd3/3, yielding

⎛ 3mN ⎞1/3 rd ⎟ =⎜ ⎝ 4π ⎠ b

(14)

Now we need to determine the conformational energy penalty (of the polymer chains) for entering the narrow capillary. To determine this entropy penalty we use the arguments of Grosberg and Khokhlov.43 (The argument is a scaling type analysis, meaning numerical prefactors are ignored.) The energy cost for an ideal chain to be confined to a tube of radius rt is based on a collisional argument. The ideal chain takes K steps to traverse from one side of the tube to the other, where rt ∼ K1/2b. Every time the chain hits the tube wall it incurs an energy penalty of kBT. A chain composed of N monomers is composed of N/K such random walks and therefore has (N/K − 1) collisions. Since K = (rt/b)2 the free energy of confinement is

Figure 13. (a) Initial system setup, where Rdi is the initial radius of the droplet and rt is the radius of the tube. (b) Intermediate stage during the absorption process with rd being the radius of the protruding droplet. (c) Droplet totally absorbed into the tube with Rde the final equilibrium radius of the protruding droplet.

FI = γLV 4πrd 2 + γSV 2πrtL

4πrt 2 1 + sin θc

(12)

In the above equations, b represents the monomer size, and L is the length that an absorbed droplet would be in contact with the tube walls.

FIIP 10263

⎞ ⎛ ⎡ ⎤2 b = mkBT ⎜⎜N ⎢ ⎥ − 1⎟⎟ ⎠ ⎝ ⎣ rt ⎦

(16)


Langmuir

Article

We can see that as rt → 0 the entropic energy diverges. Moreover, if the droplet is made up of shorter chains (N decreases and m increases but still mN is constant) the entropic energy decreases. This is as we would expect and is due to fewer collisions per chain (as the chain length decreases). The total free energy, FII, for an absorbed droplet is FII ≡ FSII + FPII. Note that this energy is valid only if the radius of gyration of individual chains which make up the droplet is comparable in size to tube radius, so that a lower bound for this energy is just FSII. The important energy is the free energy difference between an absorbed droplet and the droplet sitting at the entrance of the capillary. Thus, we define this difference as ΔF ≡ FII − FI. For convenience we scale this free energy difference by γLV4πrd2 to make it dimensionless and denote it as ΔΦ. It is given by (after some algebraic manipulation) ΔΦ ≡

=

ΔF γLV 4πrd 2

Figure 14. Surface energy difference for various droplet configurations for θc = 90° as a function of η.

2 ⎞ ⎛ 1 ⎞⎛ rt ⎞ ⎛ 1 2⎛r ⎞ ⎜ ⎟⎜ + sin θc ⎟ − ⎜ d ⎟ cos θc ⎟⎜ ⎝ 3 ⎠⎝ rd ⎠ ⎝ 1 + sin θc 3 ⎝ rt ⎠ ⎠

cos θc < 0, there is only one term favoring absorption. Figure 15 shows the free energy for a contact angle of 103.5°. (Recall the

⎞ ⎛ k T ⎞ ⎛ ⎡ b ⎤2 − 1 + ⎜⎜ B 2 ⎟⎟m⎜⎜N ⎢ ⎥ − 1⎟⎟ ⎝ γLV 4πrd ⎠ ⎝ ⎣ rt ⎦ ⎠

Note, 4πrd2 is known and equal to (4π)1/3(3mN)2/3b2. An analytically simpler, yet accurate, expression is obtained if we neglect the minus one term in the entropy. In this case we obtain ΔΦ =

⎞ 1 ⎛ 1 2 + sin θc ⎟ − η cos θc − 1 2⎜ 3 3η ⎝ 1 + sin θc ⎠ +

η2 3κ(rd /b)

(17)

where ηrd/rt and κγLVb2/(kBT). The absorbed state will be favored when ΔΦ < 0. Consider the partial wetting case first, i.e., θc ≤ π/2, and recall we are only interested in the situation where the tube diameter is smaller than the liquid droplet diameter. In this case, in the absence of a polymeric contribution (last term in eq 17) the liquid droplet will always be absorbed fully into the capillary.8 However, for polymer liquid droplets this is not the case, since the entropic term is positive and its presence favors the droplet sitting outside the tube. The case θc = π/2 is particularly instructive in this context. For large η the surface energy terms are approximately −1. However a polymeric droplet will not enter the tube when η > ηc where

ηc =

3κ(rd /b)

Figure 15. Surface energy for various droplet configurations for θc = 103.5° as a function of η.

contact angle from the MD simulations was in the nonwetting regime (≈ 103.5°).) In the absence of the polymeric term, all droplets below a radius of 6.35rt would be absorbed into the capillary. For both polymeric cases, however the critical tube radius is smaller (at 5.5rt for the 103 length chains and 5.3rt for the 106 length chain) and the critical ηc decreases with increasing chain length. In our MD simulations for the droplet of 8000 monomers, the critical ηc for short chains (N = 20) is 2.9, which for longer chains (N = 200) is 2.7. For the droplet of 16000 monomers, the critical ηc for short chains (N = 20) is 2.95, which for longer chains (N = 200) is 2.8. Thus, the theoretical results agree with our MD simulations qualitatively in that both MD simulations and theory predict (i) a decrease in critical η with longer chains and (ii) when a simple molecular fluid droplet would be absorbed, the polymeric droplet remains outside the tube. This difference can only be attributed to the entropy penalty the chains must overcome when entering a restricted domain. Finally we present results for the case of a contact angle of 135° (Figure 16). The results generally follow the previous cases. However, an important difference now is that the surface tension term in much more important. As a

(18) 1/2

This means for rt/b < ((rd/b)/3κ) the polymer droplet will not be absorbed in contrast to the nonpolymeric droplet. However, this corresponds to a very narrow tube, e.g., if rd/b ≈ 300 and κ ≈ 1 then the tube radius below which absorption ceases is 10b. Figure 14 shows the free energy for polymeric (full-lines) and nonpolymeric droplets (dashed lines) versus η for a contact angle of π/2. For this figure we use κ = 1, mN = 106, and we use eq 17 for the free energy. The nonpolymeric case, as explained above, is always fully absorbed while for only one polymer chain of length 106 the critical value of η is 13.6, while for 103 chains each of length 103 the critical η is slightly larger at 13.8. For the nonwetting case, i.e., θc > π/2, yielding 10264


Langmuir

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Figure 16. Surface energy for various droplet configurations for θc = 135° as a function of η.

result the regime where droplets will be absorbed is very small and furthermore the critical ηc for the polymeric and nonpolymeric droplets become very close. In this case we do not consider a droplet made up of chains of length 103 because their radius of gyration is smaller than the tube radius for η < 2.

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CONCLUSIONS We have presented MD simulations for absorption of polymer droplets, made up of many small chains, into narrow capillaries. The chains behave (for sufficiently short chains) like ideal chains since most of the chains are surrounded by other polymer chains. By tuning the MD interaction parameters we were able to simulate a system with a contact angle of roughly 103.5°, corresponding to a nonwetting droplet. We then simulated the absorption of droplets composed of shorter and longer chains into a narrow capillary and found in general the longer chains could not enter sufficiently narrow tubes. There is clearly an energetic barrier associated with the uptake of the droplet. But we also see chains elongating in the tube direction during the absorption which will result in entropic penalty. This could lead to an entropic barrier for absorption although it is difficult to confirm. Moreover, we also found that capillary dynamics of polymer droplets in such narrow capillaries cannot be explained using the existing models. The MD simulations and theory presented agree qualitatively. The theory also suggests for sufficiently narrow tubes, polymer droplets which wet the tube walls, may not enter the tube. This is in contrast to a simple wetting liquid which always absorbs into the tube. Our results can be tested experimentally, for example, using a nanotube or nanopore of 10 nm radius and droplets made up of polystyrene molecules with molecular weights between 104 to 105 g/mol (radius of gyration 10 to 20 nm44).

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Marmur, A. J. Colloid Interface Sci. 1988, 122, 209−219. (2) Zhmud, B. V.; Tiberg, F.; Hallstensson, K. J. Colloid Interface Sci. 2000, 228, 263−269. (3) Lucas, R. Kolloid Z. 1918, 23, 15−22. 10265


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