Odd–Even Effect of Junction Functionality on the Topology and

Article pubs.acs.org/Macromolecules

Odd−Even Effect of Junction Functionality on the Topology and Elasticity of Polymer Networks Rui Wang,† Jeremiah A. Johnson,‡ and Bradley D. Olsen*,† †

Department of Chemical Engineering and ‡Department of Chemistry, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States ABSTRACT: The junction functionality of polymer networks is a key design variable to tune the material properties through changing the network connectivity. Here, we perform a Monte Carlo simulation to study the topology and elasticity of endlinked polymer networks constructed by junctions with different functionalities. The effect of junction functionality on the topological structure of the network displays both universal features and odd−even effects. For all junction functionalities, the primary loop fraction shows a linear dependence on the dimensionless product of concentration with single chain pervaded volume. The slopes of the lines exhibit strong odd−even alternation: primary loops are formed more frequently for junctions with an odd number of functional groups. The secondary loop fraction has a maximum when plotted against the normalized primary loop fraction, where the value of the maximum also shows odd−even alternation. The topological information obtained through Monte Carlo simulation is then incorporated into the modified affine network theory and modified phantom network theory (real elastic network theory). For affine networks, the odd−even alternation in the cyclic topology leads to nonmonotonic dependence of elastic modulus on the junction functionality. For phantom networks, the effect of decreasing junction fluctuations due to a larger number of interjunction connections dominates over the changes in loop concentration as junction functionality increases, leading to monotonically increasing elastic modulus with junction functionality.

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INTRODUCTION Polymer networks are ubiquitous from commodity materials like elastomers and superadsorbents1−9 to biological tissues such as scaffolds and extracellular matrices.10−18 The threedimensional cross-linked structure determines the mechanical properties of polymer networks, the most essential factor that governs their applications as structural, medical, and responsive materials.19 However, it is notably difficult to characterize this cross-linked structure in detail: the gels cannot be molecularly dispersed in a solvent, and the detailed structures are typically not easily detected by scattering, spectroscopy, or microscopy.20 As a result, quantitatively understanding the correlation between the topology and properties of polymer networks remains one of the largest outstanding challenges in polymer science. Our fundamental knowledge about polymer networks is built upon an assumption of homogeneous loop-free treelike structures that lies at the foundation of the affine and phantom network theories.3,21−28 However, real polymer networks possess topological defects at different length scales, such as dangling ends, cyclic defects (loops of different order), and heterogeneity in cross-link density.29−34 Although dangling ends and heterogeneity in cross-link density can be largely reduced by stoichiometric end-linking of polymer precursors with multifunctional junctions,35 loops inevitably exist in all polymer networks as illustrated in Figure 1. These cyclic defects have vital effects on the mechanical strength of polymer networks.36 Quantifying the number of different loops and their © XXXX American Chemical Society

Figure 1. Schematic of end-linking bifunctional linear polymers A2 with multifunctional junctions Bf ( f varying from 3 to 8). Topological structures in the network including ideal tree, primary loop, secondary loop, and tertiary loop are illustrated by using the trifucntional junction as an example.

individual contribution to elasticity is critical to quantitatively predict the mechanical response of polymer networks, test the affine and phantom network models, and evaluate the role of trapped entanglements.27,37 Until recently, there was no way to directly count loops in polymer networks. Therefore, previous studies of cyclic defects have used percolation properties of the entire network to estimate the loop density. By using spin tree models,38−40 rate Received: September 8, 2016 Revised: January 9, 2017

A

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Macromolecules theory,41−43 and the method of moments,44−46 great theoretical efforts have been made to relate the delay of gel point compared to the Flory−Stockmayer prediction to intramolecular cross-linking. Rheology can also be used to estimate the loop density; however, this approach relies on network models whose validity have not been experimentally tested.47,48 Multiple-quantum NMR provides another tool to semiquantitatively probe cyclic defects, but it requires model functions with adjustable fitting parameters and cannot discriminate dangling chains and primary loops.49,50 Recently, we reported variations of “network disassembly spectrometry” (NDS), which are applicable to a wide range of polymers and provides the first experimental approach to directly count the number of primary loops in polymer networks.51−53 Combining NDS with rheological measurement shows significant deviation of the elastic modulus of real polymer networks from the ideal network models even though elastically inactive primary loops have been subtracted, demonstrating the role of higher order loops that cannot yet be experimentally measured.36 The existence of higher order loops has also been shown using model Tetra-PEG gels, in which primary loops are eliminated through the careful choice of precursors.54 Motivated by the availability of this new experimental data,51−53 we recently developed a kinetic graph theory based on previous theoretical work42,43,55,56 to study the cyclic topology of networks prepared via end-linking of bifunctional polymer precursors and trifunctional small molecule junctions.57 The primary loop fraction predicted by the theory shows excellent agreement with the published NDS experimental data without any fitting parameters. This theory reveals that the entire cyclic topology can be described by the primary loop fraction; the relative populations of loops of different orders are not freely adjustable. The kinetic graph theory was also confirmed by Monte Carlo simulations, showing quantitative agreement for low order loops. To bridge the network topology and the gel elasticity, we also developed a real elastic network theory (RENT), which systematically incorporates the impacts of different cyclic defects into the phantom network theory.36 The theory shows a vital negative effect of small loops (primary loops and secondary loops) on the linear elastic modulus of polymer networks, and it can quantitatively predict the modulus without any fitting parameters over a broad range of concentration. The functionality of junctions, f, is a key variable to tune the material properties through changing the network connectivity. However, systematic computational and theoretical study of the effect of junction functionality on the topology and elasticity of polymer networks is lacking. Although trifunctional (f = 3) and tetrafunctional (f = 4) junctions are the most typically used cross-linkers in chemical gels,35,53,58 there is recent interest in using metal−organic cages as the junction whose functionality can be much larger than 4 (even up to 24 for the case of the “Fujita cage”).59,60 Junctions with functionalities higher than 4 are also common in bio-inspired hydrogels with protein-based cross-links.61 According to the phantom network theory, the elastic modulus of ideal loop-free networks is proportional to 1− f/2, indicating that increasing junction functionality creates more intermolecular connections between the junctions to reduce junction fluctuations. On the other hand, increasing junction functionality also increases the probability of intramolecular cross-linking, which produces more cyclic defects not captured in the theory. Therefore, the effect of junction functionality on the network topology should result from a

competition between these two aspects. Given the same number of polymer strands, it is also interesting to understand the possibility of tuning the elasticity of polymer networks by changing the network connectivity due to different junction functionalities. In this work, Monte Carlo simulation is used to study the cyclic topology of polymer networks prepared via end-linking reactions. Molecular simulation provides an important tool for the study of both higher order topological structures and higher junction functionalities where the size of reaction networks required to capture structure formation makes the kinetic graph theory impractical.37,50,63−65 Furthermore, the Monte Carlo algorithm has already been experimentally validated for low junction functionalities using direct measurements of primary loops.51−53 Using junctions with different functionalities, the dependence of cyclic defects on the preparation condition as well as the inherent relations between different orders of loop structures is investigated. The topological information obtained through Monte Carlo simulation is then incorporated into the modified affine network theory and modified phantom network theory (real elastic network theory) to address the effect of junction functionality on the elastic modulus of polymer networks.

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MODEL AND SIMULATION ALGORITHM As illustrated in Figure 1, the polymer networks considered are formed via end-linking of bifunctional linear polymers A2 and branched multifunctional junctions Bf, where f (f > 2) is the number of functional groups (i.e., functionality) on the junction. Nishi et al. show that the end-linking process is not diffusion-limited but reaction-limited over a wide range of concentrations,62 which allows simulation based on a purely topological perspective: the spatial information on the polymers and junctions can be ignored, and the simulation can be simplified by only tracking the topological distance between reactive groups. The multifunctional junction Bf is modeled as a small monomer; the distance between two adjacent functional groups within the same junction is set to be 0. The bifunctional linear polymer A2 is modeled as a dispersity 1.0 flexible Gaussian chain, where two A functional groups are located at the respective ends of the chain. The probability distribution P(r) of the spatial distance between these two A functional groups is then given as27 P(r ) =

⎛ 3r 2 ⎞ ⎛ 3 ⎞ ⎜ ⎟ exp⎜ − ⎟ 2 ⎝ 2πR ⎠ ⎝ 2R2 ⎠

(1)

with R the root-mean-square end-to-end distance of the Gaussian chain. R = N1/2b, which can be calculated from the degree of polymerization N and Kuhn length b for a given type of polymer. The topological distance between any two functional groups in the system is defined as the number of polymer chains in the shortest topological pathway that connects these two groups. Based on this definition, the distance between two A functional groups in the same polymer chain is straightforwardly 1, and that between any two unconnected groups is infinity. If a pair of A and B functional groups reacts, their distance is set to be 0; the topological information for all the groups connected to these reacted A and B is also updated after each reaction event. All A functional groups (or B functional groups) are assumed to have equal reactivity during the entire end-linking process. The model presented here can be easily generalized to polymers with other B

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Figure 2. (a) Primary loop fraction x1,f versus the dimensionless variable cR3 (c is the polymer concentration and R is the root-mean-square end-toend distance of the polymer chain) as f varies from 3 to 8. (b) Linear relation between xs1,f/x1,f − 1 and cR3. (c) The log−log plot of x1,f/xs1,f with respect to cR3. (d) The slope kf of linear dependence of xs1,f/x1,f + x1,f on cR3 as a function of the junction functionality f.

analysis of network topology. This reaction creates a new pathway between all groups that are connected to these A and B groups. Depending on whether this new pathway shortens the topological distance, the update of dij will be accepted or not. For example, if group i is connected to the reacted group A with distance diA and group j is connected with the reacted group B with distance diB, the new topological distance dnew ij between groups i and j after the reaction is given as

chain statistics, such as semiflexible polymers, and with disperse chain lengths, as well as functional groups of unequal reactivity. To track the formation of the network topology through molecular connections, Monte Carlo simulation is used following the general algorithm developed by Stepto63 and then adapted according to Zhou et al.51,52 Based on the desired polymer concentration c, NA A2 polymers (with 2NA A functional groups) and NB Bf junctions (with f NB B functional groups) are initially put in the system of volume V (c = NA/V). Networks are constructed by sequentially selecting a pair of unreacted A−B groups at each step with a probability PAB reflecting the topological connection between these two groups

PAB =

1 V

+

(

3 2πR2dAB

dijnew = min(dijold , di A + dj B)

where dijold is their distance before the reaction. For intermolecular reaction (i.e., the reaction occurring between two unconnected subnetworks), the two subnetworks infinitely far away from each other are connected together; therefore, the old topological distance must always be replaced by the new one in this reaction. At each reaction step, the available unreacted A−B pairs decrease and the topological distance matrix dij is changed; the denominator of the selection probability in eq 2 is reevaluated. The process is repeated until either A functional groups or B functional groups are fully consumed. The fractions of different topological defects with respect to the total number of polymer chains are recorded. Dangling chains are the polymers with one of their two A functional groups having not been connected to any B functional group. Primary loops are when both A groups belonging to the same polymer chain connect to the same junction (two adjacent B groups). Similarly, higher order cyclic structures, such as secondary and tertiary loops, can also be identified based on the connectivity as illustrated in Figure 1b. In the simulation, about 15000 A functional groups (7500 A2 polymers) and 15000 B functional groups (15000/f Bf junctions) are initially put in the system at various

3/2

)

⎡ ⎛ 3 ⎞3/2 ⎤ 1 ∑ij ⎢ V + ⎜ 2 ⎟ ⎥ ⎝ 2πR dij ⎠ ⎥⎦ ⎢⎣

(3)

(2)

where the summations are taken over all unreacted A−B pairs. dij is the topological distance (defined by the number of A2 chains in the shortest topological pathway) connecting the ith A group and the jth B group. dij = ∞ if the two groups are not connected. The selection probability PAB has two contributions: the term 1/V is the pairwise concentration of any pair regardless of their connectivity and the surrounding networks; the additional term (3/2πR2dij)3/2 reflects the internal concentration if the two groups are connected to the same network. The scaling of this term comes from the statistics of flexible Gaussian chains for loop closing as described by eq 1. The topological information about the network is updated after the selected A−B pair reacts. The connection between the two selected groups is also recorded, which will be used in the C

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Macromolecules concentrations. The number of functional groups is chosen to be large enough so that the loop fractions are not sensitive to the system size. Ten parallel simulations were run for each set of parameters to get the average fraction of topological defects. The Monte Carlo simulation algorithm presented here can be applied to polymer networks formed by any mixing ratio of bifunctional polymers A2 and multifunctional junctions Bf. In this work, we focus on the stoichiometric reactions where equal amounts of A functional and B functional groups are carried out in the simulation. As full conversion of the stoichiometric reactants is achieved, the presence of dangling ends (chains and junctions) can be eliminated, which enables direct study of the cyclic defects in the polymer network.

To better illustrate the decay rate of the primary loop fraction and reveal the intrinsic dependence on the preparation condition of polymer networks, the reciprocal of primary loop fraction (normalized by its saturated value xs1,f) is plotted against cR3 as shown in Figure 2b. 1/x1,f exhibits a linear relation with cR3, similar to the behavior of a Langmiur adsorption isotherm. This linear relation holds in the entire range of junction functionalities studied in this work, leading to the following analytical expression to approximate the primary loop fraction:

RESULTS AND DISCUSSION Polymer networks constructed by multifunctional junctions with f from 3 to 8 are considered, which covers the range of junction functionality used in most experimental systems. The loop fractions xn,f are defined as the number of polymer chains contained in all nth-order loops divided by the total number of polymer chains in the network. Given that the density of polymers is held constant as functionality is varied, this definition facilitates a fair comparison of the cyclic topology between networks constructed by different junction functionalities as discussed in the following sections. Primary Loops. The primary loop fraction is a uniquely identified topological feature that provides important information to characterize polymer networks. As demonstrated in our previous work, the dimensionless loop fractions for all orders of loop xn,f are uniquely determined by the dimensionless variable cR3 (c is the polymer concentration and R is the root-meansquare end-to-end distance of the polymer chain), which characterizes the ratio between the intramolecular distance and the intermolecular distance. The pervaded volume R3 can be expressed as R3 = (M/m)3/2b3 in terms of molar mass of polymer A2 (M), Kuhn length (b), and the molar mass of the Kuhn monomer (m), which are either experimentally measurable or reported in the literature.27 cR3 is also proportional to c/c*, where c* is the overlapping concentration in polymer solutions that can be measured in experiments and related to R3 as c* ∼ 1/R3. Varying f at a fixed value of cR3 reflects the effect of junction functionality on the construction of the network while maintaining the same preparation condition in terms of linear polymers. As shown in Figure 2a, the primary loop fraction x1,f decays monotonically as cR3 increases for all different values of f explored. As cR3 goes to 0, x1,f approaches the saturated primary loop fraction xs1,f in the full-loop limit, where the polymer networks are sols composed of individual junctions or pairs of junctions with predominantly primary loops. The saturated primary loop fraction shows an odd−even effect in f: if f is an even number, f/2 primary loops will share one junction in this limiting state; if f is an odd number, (f − 1)/2 primary loops will share one junction, and the remaining unreacted functional group in this junction will participate in a bridge. xs1,f is thus given by

where the coefficient kf is the slope of the line. For cR ≪ 1, x1,f ∼ (cR3)0, indicating the high dilution regime where the maximum number of loops are formed. For cR3 ≫ 1, x1,f ∼ (cR3)−1; primary loops are rare and separated by ideal bridging structures. The network, most probably a percolated gel, can be envisioned as an “ideal loop gas”. This scaling behavior of x1,f with respect to cR3 is clearly illustrated in a log−log plot as shown in Figure 2c. The slope kf reflects the sensitivity of the primary loop fraction to the junction functionality. A larger value of kf means that the primary loop fraction is reduced faster by increasing the initial concentration or the chain length of the polymers. The dependence of kf on the junction functionality f is plotted in Figure 2d. It is surprising that the decrease of kf with f is not monotonic but exhibits a strong odd−even alternation: kf for even f are relatively larger than those for odd f. The larger the junction functionality, the smaller the relative difference between the even and odd groups. In general, kf decreases as f increases, which is expected because larger junction functionality provides more available functional groups to produce intramolecular loops, meaning that loops can persist to higher cR3. In contrast, previous studies limited to trifunctional ( f = 3) and tetrafunctional ( f = 4) systems incorrectly indicated an opposite trend that the normalized primary loop fraction x1,f /xs1,f is reduced with increasing f since kf=4 > kf=3. Therefore, a systematic study of a wide range of f is necessary to avoid the misleading that cyclic defects in the polymer network can be reduced by continuing to increase the junction functionality. This odd−even effect is explained based on the topological constraints on polymer networks: the possible ways to allocate chains between loops and bridges on odd and even junctions. An intermolecular bridge occupies one functional group in a junction, whereas an intramolecular primary loop requires two functional groups. As cR3 increases, the number of bridges will become increasingly large. If one bridge forms on a junction with odd f, the remaining even functional groups may still pair to primary loops. On the contrary, if one bridge forms on a junction with even f, at least one other bridge has to be formed on the same junction to satisfy the restriction of parity no matter how large the probability to form a loop. In other words, junctions with even f that have already been connected to the network through a bridge can only provide an odd number of available functional groups, so bridges on these junctions must occur in pairs. Therefore, networks with even f produce fewer primary loops compared to those with an odd f. While odd−even effects have been observed before, they have not been previously predicted in gels. Odd−even effects have usually been reported in the dependence of physical properties, such as melting point, solubility, elastic modulus,

x1, f =

■

⎧1 for even f x1,s f = ⎨ ⎩(f − 1)/f for odd f

1 + k f cR3

(5) 3

⎪ ⎪

x1,s f

(4)

3

As cR increases, Figure 2a also shows that the decay rate of x1,f depends on the junction functionality, which results in crossovers between the x1,f curves at low values of cR3 for different values of f. D

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Figure 3. (a) Secondary loop fraction x2,f and (b) the tertiary loop fraction x3,f versus the dimensionless variable cR3 as the value of f varies from 3 to 8. (c) x2,f and (d) x3,f versus the normalized primary loop fraction x1,f/xs1,f. (e) The maximum value of x2,f and x3,f as a function of the junction functionality f.70 (f) The value of x1,f/xs1,f at the maximum of x2,f and x3,f as a function of f.

reflects the competition between different topological structures due to the saturation of junction functionality. Primary loops dominate for cR3 ≪ 1, whereas intermolecular bridges dominate for cR3 ≫ 1. x2,f and x3,f approach 0 in these two limiting regions and display a maximum for cR3 of order 1 (approximately at cR3 = 0.5). To better illustrate the inherent relation between different cyclic structures, the secondary loop fraction x2,f and the tertiary loop fraction x3,f are plotted as a function of the primary loop fraction as shown in Figures 3c and 3d, respectively. To rescale the primary loop fraction for different values of f to the same range, x1,f is normalized by its saturated value xs1,f. With this normalization, the plots of x2,f for different f show universal features: the maxima of x2,f for different f appear approximately at the same position around x1,f/xs1,f = 0.5. As shown in Figure 3e, it is surprising that the maximum value of x2,f does not increase monotonically with f, but rather displays an odd−even alternation, which is similar to the decay rate of the primary loop fraction kf. Junctions with even functionality have

isotropic−nematic transition, and self-assembly of monolayers, on the number of CH 2 units in alkanes and their derivatives.66−69 In many systems this odd−even alternation is explained by the “packing” of carbon skeleton along the principal axis. Here, it is clear that a similar odd−even effect occurs not only in the saturation level of loops, which is clearly evident based on connectivity patterns at infinite dilution, but also in the concentration dependence of loops as a function of junction functionality due to the topological constraints on polymer networks, which has not been reported before. Higher-Order Loops. While primary loops are completely elastically ineffective, higher-order loops partially contribute to the elasticity of polymer networks. The higher-order loops cannot yet be experimentally quantified, which makes computer simulation an essential tool to study these topological defects. The fractions of secondary loop x2,f and tertiary loop x3,f for different values of f are shown in Figures 3a and 3b. Unlike the monotonic decay of the primary loop fraction, x2,f and x3,f present a nonmonotonic change with respect to cR3, which E

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Macromolecules comparatively larger maximum values of x2,f than the odd group. The difference between the even group and the odd group decreases as f increases. This trend mirrors the trend in the primary loops, consistent with the fact that for low cR3 the decrease in primary loops enables an increase in secondary loops. For functionalities where the primary loops decrease faster, the secondary loops increase faster. The odd−even effect for the maximum value of the tertiary loop fraction is much weaker compared to the effect for primary and secondary loops. As shown in Figure 3e, the maximum of x3,f increases monotonically with increasing f, where only a slight difference between the odd group and the even group can be observed. As f increases, the functional groups available to form the higher order loops increase faster than those available for the lower order loops, which compensates the impact from the saturation of junction functionality that leads to the odd−even alternation. Therefore, the strong odd−even alternation observed in the primary loop and secondary loop becomes insignificant in tertiary and higher order loops. While the value of x1,f/xs1,f corresponding to the maximum of x2,f remains around 0.5 for different f, the position of the maximum of x3,f surprisingly shows odd−even alternation as shown in Figure 3f. The underlying mechanism that explains the different behaviors between the secondary loop and the tertiary loop is unclear. The Monte Carlo simulation algorithm used in this work has been experimentally validated by comparison to experimental measurements of loop densities for trifuctional and tetrafunctional networks,51−53 with the key limitation that the simulations are applied in the reaction-limited regime. By focusing on the topological perspective of the network while neglecting the spatial information on the polymer segments and junctions, this simulation can capture larger numbers of chains than spatially dependent methodologies,37,71 resulting in a more accurate description of the network structure. As long as the process is not diffusion-limited, the end-linking in the Monte Carlo simulation used here is allowed to take place between any pair of unreacted A−B groups in the system with a weighted probability, instead of being restricted to the neighboring reactive groups.64,65 The assumption of reactionlimited end-linking process has been validated by experiments in a wide range of concentrations;62 however, it should be noted that the real space simulation is necessary if the endlinking takes place in the diffusion-limited regime. Compared to the graph-based theories,29,30,41−43,55−57 this Monte Carlo simulation is able to capture a larger set of cyclic topologies, which is crucial for exploring the intrinsic relation between different orders of loops as illustrated in Figure 3. Elasticity. The odd−even alternation of network topology with changing junction functionality will have a significant effect on the mechanical response of polymer gels. Classical network theories, such as affine network theory and phantom network theory, are usually used to connect the network topology to the gel elasticity; however, the range of applicability of each theory is not yet clear. For example, a transition from phantom to affine network model has been observed by either increasing the polymer concentration or the chain length.54 Therefore, the effect of junction functionality on gel elasticity will be separately explored in both affine and phantom network models. In the affine network model, the relative deformation of each network strand is the same as the macroscopic relative deformation imposed on the entire network. Polymer strands in primary loops are completely elastically inactive whereas

strands in higher order loops are assumed to be totally elastically effective in the affine model. By subtracting the wasted strands, the shear modulus Gaf′ can thus be calculated as Gaf′ ν0kT

= 1 − Aaff x1, f

(6)

with ⎧ 3 for f = 3 ⎪ Aaff = ⎨ 2 for f = 4 ⎪ ⎩1 for f ≥ 5

v0 is the total number density of polymer strands (including both loop and nonloop). The difference of the coefficient Aaff between different f is due to the saturation of junction functionality. For primary loops in the trifunctional ( f = 3) system, the loop and the strand connected to the loop are elastically ineffective, and the two strands connected to the elastically ineffective strands rejoin to a single elastically effective strand. This yields a reduction of three elastically effective chains for each loop. In the tetrafunctional (f = 4) system, the loop is elastically ineffective, and the two strands connected to the loop rejoin to a single elastically effective strand. This yields a reduction of two elastically effective chains per loop. The modified affine network theory (eq 6) assumes that different cyclic defects are isolated and independent of each other; there is no cooperative effect between different loops. Therefore, the gel can be envisioned as an “ideal loop gas”, which enables an isolated treatment of each cyclic defects. To satisfy the “ideal loop gas” assumption, the range of 1/(cR3) is chosen to guarantee that the primary loop fraction is small enough (