Giant Casimir Non-Equilibrium Forces Drive Coil to Globule Transition

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Chemical and Dynamical Processes in Solution; Polymers, Glasses, and Soft Matter

Giant Casimir Non-Equilibrium Forces Drive Coil to Globule Transition in Polymers Himadri S. Samanta, Mauro Lorenzo Mugnai, Theodore R. Kirkpatrick, and Devarajan Thirumalai J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b00695 • Publication Date (Web): 08 May 2019 Downloaded from http://pubs.acs.org on May 10, 2019

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The Journal of Physical Chemistry Letters

Giant Casimir Non-Equilibrium Forces Drive Coil to Globule Transition in Polymers Himadri S. Samanta,† Mauro L. Mugnai,† T. R. Kirkpatrick,‡ and D. Thirumalai∗,† †Department of Chemistry, University of Texas at Austin, TX 78712 ‡Institute For Physical Science and Technology, University of Maryland, College Park, MD 20742 E-mail: [email protected]

Abstract We develop a theory to probe the effect of non-equilibrium fluctuation-induced forces on the size of a polymer confined between two horizontal thermally-conductive plates subject to a constant temperature gradient, ∇T . We assume that (a) the solvent is good and (b) the distance between the plates is large so that in the absence of a thermal gradient the polymer is a coil whose size scales with the number of monomers 5

as N ν , with ν ≈ 0.6. We find that above a critical temperature gradient, ∇Tc ∼ N − 4 , favorable attractive monomer-monomer interaction due to Giant Casimir Force (GCF) overcomes the chain conformational entropy, resulting in a coil-globule transition. Our predictions can be verified using light-scattering experiments with polymers, such as polystyrene or polyisoprene in organic solvents in which the GCF is attractive.

May 7, 2019

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HOT

∇Tc ≈ N −5/4 COLD

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Long-ranged interaction forces in nature are often caused by fluctuations of some physical entity in restricted geometries. Well-known examples include the Casimir interaction, which is the macroscopic manifestation of quantum fluctuations of the electromagnetic field. The Casimir interaction force, fEM , arises due to the changes in the vacuum energy density, in the presence of two neutral perfectly conducting boundaries. 1,2 These ideas were further generalized by Lifshitz for dielectric material characterized by frequency-dependent dielectric permittivity. 3,4 Subsequently, Fisher and de Gennes remarked that a similar effect could emerge in condensed phases as well. 5 For confined critical systems, such as a fluid near the liquid-gas critical point, a binary liquid near the consolute point, or liquid He4 near λ transition, critical fluctuations of the order parameter generate long range forces between the confining walls, denoted by critical Casimir forces, fc . 6 Of particular interest here are the long-ranged correlations in confined fluids subject to a temperature or concentration gradient. 7,8 A physical reason is that the intensity of the NE fluctuations diverges for small wave number k as k −4 , while the intensity of critical fluctuations only diverges as k −2 . Hence, the non-equilibrium fluctuations are much more dramatic than critical fluctuations. The magnitude of the effects are so spectacular that they can have been measured using standard light scattering experiments. 9 A recent series of studies have investigated the nature of the fluctuation-induced force generated between two parallel plates in a fluid with a temperature gradient ∇T . 10–12 In this case, the extremely long-ranged correlations induce forces (referred to as Giant Casimir force) whose magnitudes are considerably larger than fEM and fc . The estimate for the order-ofmagnitude of NE Casimir pressure is presented elsewhere. 10–12 This force (fN E ) results from thermal fluctuations in a fluid in non-equilibrium (NE) steady state. At large distances, fN E > fc > fEM . 10 For completeness, we point out that that fluctuation-induced forces in a fluid mixture near an equilibrium critical point, which are smaller than fN E have also been studied. 13,14 Here, we explore the possibility that fN E could have profound consequences on

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the size of a polymer confined between two plates (Fig.1) that is subject to a constant ∇T . Temperature gradient-induced effects on colloids and polymers have been previously investigated in a variety of contexts. Examples include separation of macromolecules in organic solvents, 15 crowding of nucleotides, 16,17 colloidal accumulation in micro-fluid, 18 phase separation in a miscible polymer solution, 19 structural evolution in directional crystallization of polymers, 20 and thermophoresis. 21 Here, we investigate the effect of the non-equilibrium giant Casimir force (GCF), 10 fN E , on the size of a polymer. We develop a theory to assess the effects of GCF on a self-avoiding homopolymer with N monomers of size a0 confined between two parallel thermally conducting plates separated by a distance L subject to a uniform temperature gradient, ∇T (Fig. 1). We assume that the fluid is a good solvent for the polymer, implying that the radius of gyration of the polymer is Rg ∼ a0 N ν with ν ≈ 0.6 if ∇T = 0; furthermore, we assume that

L Rg

1, which implies that the polymer behaves

as a Flory random coil. We argue that the presence of the GCF induces an intramolecular attraction, which we predict to be sufficiently strong to overcome the conformational entropy of the polymer, thus inducing a genuine NE coil-to-globule transition. Such a transition occurs in equilibrium (∇T = 0) only when the solvent quality is changed from good to poor. 22 For a fixed N , we predict that there is a critical temperature gradient, ∇Tc , above which the polymer undergoes a coil → globule transition. At ∇Tc the monomer-solvent energetics

Hot Plate: T+𝛅T/2 z y

r(s)

x

L

Cold Plate: T-𝛅T/2

Figure 1: A schematic of a homopolymer in a fluid (good solvent) confined between two parallel thermally conducting plates located at z = 0 and z = L. The temperature difference between the plates is δT and ∇T = δT L .

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compensates for the monomer-monomer interactions, thus defining the NE Θ-point. Theory: Consider a polymer chain (Fig. 1) dissolved in a fluid confined between two horizontal thermal conducting plates, located at z = 0 and z = L with stationary temperature gradient ∇T . Convection effects can be avoided by heating the fluid from above as indicated in Fig. 1. We start with the Edwards Hamiltonian for a polymer chain, 3kB T HE = 2a20

ZN

∂r ∂s

2 ds + kB T V(r(s)),

(1)

0

where, r(s) is the position of the monomer s (Fig. 1), a0 is the monomer size, and N is the number of monomers. The first term in Eq. (1) accounts for chain connectivity, and the secRN RN 0 ))2 ]. ond term represents volume interactions given by V(r(s)) = 2(2πav2 )3/2 ds ds0 exp[− (r(s)−r(s 2 2a 0

0

0

0

We consider a long polymer whose size, Rg , is much less than L. In this limit, it is known that Rg ≈ a0 N ν with ν ≈ 0.6 provided v > 0 i.e., the polymer is in a good solvent. In the presence of ∇T , the interaction between two monomers at r1 = r(s1 ) and r2 = r(s2 ), is altered due to fluctuation-induced non-equilibrium forces. The additional effective pair potential due to GCF, V (r1 , r2 ), could be attractive or repulsive depending on the thermodynamic properties of the fluid. In a one component fluid, one encounters a heat mode causing temperature fluctuations, sound modes associated with pressure fluctuations and a viscous mode causing velocity fluctuations. 23,24 A fluid layer subjected to a temperature gradient ∇T is described by two coupled fluctuating hydrodynamic equations of temperature and velocity fluctuations. 25 The temperature gradient causes a coupling between the temperature fluctuations and the velocity fluctuations, i.e coupling between heat and viscous mode. 8 The NE fluctuations, arising because of the coupling between the heat and the viscous modes, are long ranged and nonlocal. In normal liquids, sound modes are fast propagating modes associated with pressure fluctuations, while the heat mode is a slow, diffusive one. NE fluctuations are unaffected by sound modes. We consider the pressure as a function of fluctuating mass density δρ and

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energy density δe. To deal with slow diffusive temperature fluctuations, we can neglect the fast propagating sound modes, and hence, the linear fluctuation contribution to the pressure in the Taylor expansion in δρ and δe is zero. We only retain the terms quadratic in the fluctuations of mass and energy densities. 10–12 For a quiescent fluid, in the presence of a uniform temperature gradient ∇T , the nonequilibrium contribution to the intensity of the temperature fluctuations varies with the wave number k, diverging as k −4 in the limit k → 0, as predicted theoretically 7,26–28 and confirmed experimentally. 9,29–35 As a consequence, it was recently shown that the long-range NE fluctuations induce significant Casimir-like forces in fluids, in the presence of temperature gradient. We propose that the effect of these GCF could be observed by monitoring the size of a polymer in solution. The magnitude of the predicted effect would be a macroscopic realization of the non-equilibrium GCF. The L-dependent NE fluctuation contribution to pressure, PN E (L), is given by, 10–12 PN E (L) = kB T A L

∇T T

2 = kB T

A δT 2 , L T

(2)

with CP T (γ − 1) 1 ∂CP 1 ∂α A= 1− + 96πDT (ν + DT ) αCp ∂T P α2 ∂T P where δT is the temperature difference between the two plates (Fig. 1), CP is the isobaric specific heat capacity, DT is the thermal diffusivity, γ is the ratio of isobaric and isochoric heat capacities, and α is the thermal expansion coefficient. For a fixed ∇T , the NE pressure grows with increasing L. Thus, for a given ∇T the giant NE Casimir pressure varies as L−1 , which is much longer ranged than the fEM ∼ L−4 dependence for electromagnetic Casimir forces, or fc ∼ L−3 dependence for the critical Casimir forces. 5 Note that Eq. 2 implies there is a long-ranged NE repulsive force between the confining walls if A > 0. This suggest that an object placed between the plates will feel a compression force. To assess the effect of non-equilibrium fluctuations, we imagine an experiment in which

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a large polymer with Rg /L > Rg . While the former condition can be enforced experimentally by reducing the concentration of the polymer, the latter depends on the chemical nature of the polymer and the solvent through the Soret coefficient. Calculation Details: In order to calculate the Rg of the polymer subject to GCF, we use an approximate method introduced by Edwards and Singh (ES). 38 The ES method is a variational type calculation that represents the exact Hamiltonian by a Gaussian chain. The effective monomer size in the variational Hamiltonian is determined as follows. Consider a virtual chain without excluded volume interactions, with the radius of gyration hRg2 i = N a2 /6, 38 described by the Hamiltonian

3kB T Hv = 2a2

ZN

∂r ∂s

2 ds,

(4)

0

where a is the effective monomer size. We split the deviation W between the virtual chain Hamiltonian and the exact Hamiltonian as,

HE − Hv = kB T W = kB T (W1 + W2 ),

(5)

where 3 W1 = 2

1 1 − 2 2 a0 a

W2 = V(r(s)) +

ZN

0 N X

∂r ∂s

2 ds,

(6)

V (r(s) − r(s0 )).

s,s0 =0

The term V (r(s) − r(s0 )) in Eq. (6) is the contribution from the GCF, and is obtained by integrating F12 in Eq. (3) with respect to the z-variable. The radius of gyration is R 2 −H /k T −W RN 2 2 −W i v B e r e δr 1 v 2 2 Rg = N hr (s)ids, with the average being hr (s)i = R e−Hv /kB T e−W δr = hr he(s)e , where −W i v 0

h· · · iv denotes the average over Hv . Assuming that the deviation W is “small,” we can calculate the average Rg to first 8


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order in W. If we choose the effective monomer size a in Hv such that the first order correction vanishes, then the size of the chain is, hRg2 i = N a2 /6. This is an estimate of the exact hRg2 i, and is an approximation as we have neglected W 2 and higher powers of W. However, following the analysis in

38

we can assume that inclusion of higher order terms

merely renormalizes the coefficients of the dependence of Rg on N and ∇T without altering the essential qualitative results. Thus, in the ES theory, we find a by setting the first order correction to zero. The resulting self-consistent manner for equation for a is: 38

1 1 − 2 = 2 a0 a

1 N

RN 2 [hr (s)iv hW2 iv − hr2 (s)W2 iv ]ds 0 . R a2 N 2 (s)i ds hr v N 0 1 N

Calculating the averages in the Fourier space (rn = P Rg2 = 2 h|rn 2 |i), we obtain

RN

πns N

cos

1

(7)

r(s)ds; r(s) = 2

∞ P

cos

n=1

πns N

n

1 1 − 2 = 2 a0 a

0

where, C1ss = ∞ P

3a5 π

√ 2N ∞ P 5/2 n=1

ZN

∞ P 1 n2

n=1

4a20 N 1−cos[nπ(s−s0 )/N ] 2 , ∞ P n4 1 9π n2 n=1

ZN ds

0

"

0

C ss ds0 v ss10 5/2 − A (C )

∇T T

2

# ss0

C2

(8)

0

0

0

)/N ] 2 1−cos[nπ(s−s , C ss = n4

2N 3π 2

∞ P n=1

1−cos[nπ(s−s0 )/N ] n2

+

a20 , a2

0

C2ss =

and v = 43 πa30 . We constrain the center of mass of the polymer

n=1

to remain at fixed position, r0 = 0, in order to get rid of the center of mass difusion. The best estimate for the effective monomer size a can be obtained using Eq. (8). The Θ-point signals the transition from a coil to a globule. We use the definition of the Θ-point to assess the condition for collapse in terms of temperature gradient. At the Θ-point, the 0 2 ss0 C ss v-term ( v (C ss10 )5/2 ) should exactly balance the GCF term (A ∇T C2 ). Since at the ΘT point the dimensions of the chain is ideal, implying a = a0 , we can substitute this value for a in the v- and the GCF terms, and equate the two . The result yields an expression for the Θ-point as a function of ∇T . Thus, from Eq. (8) the critical temperature gradient at which

9


rn ;


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two body repulsion (v-term) equals two body interaction (Casimir-term) is, RN ∇T 2 c

T

=

ds

RN

0

0

A

RN 0

0

C ss

ds0 v (C ss10 )5/2

ds

RN

.

(9)

ss0

ds0 C2

0

The numerator in Eq. (9) is a consequence of chain connectivity, and the denominator encodes the fluctuation induced effect, determining the extent to which the size of the polymer changes with temperature gradient. Clearly, ∇Tc is determined by the fluid properties through A. The results in Eq. (8) can be used to obtain the dependence of ∇Tc on N . Scal0

0

0

ing n by N , it can be shown that C1ss ∼ N . Similarly, C2ss ∼ N and C ss ∼ N . From these results it follows (see the Supplementary Information for an alternative derivation) that 5

∇Tc ∼ N − 4 .

(10)

Practical considerations: To estimate the critical temperature gradient ∇Tc , implied by Eq. (9), we consider the example of fluid neopentane, for which accurate light scattering experiments of the non-equilibrium temperature fluctuations are available. Using available data for the thermodynamic and transport properties for neopentane [described in the Supplementary Information (SI)], we calculated the numerical value ∇Tc using Eq. (9) at the Θ transition. As an example consider the parameters given in Table S2 in the SI. The predicted value (Eq. 10) for ∇Tc at 433K for neopentane ranges from ≈ (0.55 − 0.13) K · µm−1 for N between 10,000 to 30,000 (see the square points in Fig. 2). We ought to point out that in our model the largest energy of interaction between two monomers due to the GCF is approximately V (aN ν ) ≈ 0.07 pN · nm for a polystyrene polymer in good solvent (neopentane) approaching the critical gradient ∇Tc ≈ 0.55 K/µm at T = 433K, with N = 104 , ν = 0.588, and assuming a ≈ a0 . This value is two orders of magnitude smaller than the thermal energy (≈ 6.0 pN · nm). The coefficient A of GCF term is positive for the fluid neopentane, implying that a 10


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2

● T=432K; ρ=4.1745 mol/l ■ T=433K; ρ=3.9401 mol/l

●

1

∇Tc (K/μm)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


∇Tc ∼N-5/4 ∇Tc ∼N-5/4

● ●

0.5■

● ●

■

0.2

■ ■ ■

0.1 10 000

15 000 20 000

30 000

50 000

N

Figure 2: The critical temperature gradient ∇Tc decreases with N as N −5/4 . Circular and rectagular points are the predicted values for ∇Tc for neopentane fluid at temperature 432K and 433K respectively, for different polymer chains. The calculations are done for polystyrene for which a0 , the monomer size, is ≈ 17˚ A. 39

polymer for which neopentane is a bad solvent (such as polyisoprene or polystyrene) is predicted to undergo a coil to globule transition when ∇T exceeds ∇Tc . The GCF term is very sensitive to the thermodynamic properties of the relevant fluid. For toluene, the coefficient A of GCF is negative for temperature below 310K at 26 MPa pressure, implying fluctuation-induced interactions between monomers are repulsive (see the SI for the values for toluene). In this case, the chain is in a good solvent even if ∇T 6= 0, implying that there ought to be no coil-globule transition for any value of ∇T . Above 310K temperature, the coefficient A of GCF term is positive, implying fluctuation induced interactions between monomers are attractive and the coil-globule transition would occur when ∇T exceeds ∇Tc . These spectacularly contrasting predictions can be verified using currently available techniques with standard polymers (polystyrene or polyisoprene) in organic solvents. The experimental test of our predictions could present technical challenges. Both the synthesis of large polymers, and the setup of the experimental apparatus capable of holding the necessary temperature gradient require careful planning. However, given the advanced status of current experimental capabilities, 40 and the size of the predicted changes in Rg , we expect that it may be possible to find a compromising choice of polymer size and magnitude of the thermal gradient, which would allow a direct testing of our predictions. Final Comments: (i) We have shown that non-equilibrium giant Casimir forces between 11



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monomers in a homopolymer in a temperature gradient induce a coil to globule transition in a polymer that exists in a swollen random coil state in the absence of the temperature gradient. The interactions leading to the predicted dramatic transition is due to attractive long ranged interactions between the monomers due thermal fluctuations in non-equilibrium. The fluctuation-induced forces could be attractive or repulsive depending on the thermodynamic properties of the relevant fluid. (ii) Our theory predicts that above a critical value of the temperature gradient, monomer-monomer attractive interactions will overcome the chain conformational entropy, inducing a coil to globule transition in certain solvents. The easiest experiment to imagine is a polymer in a custom-made cell in a fluid with the system initially in thermal equilibrium. A vertical temperature difference ∆T across the cell is then imposed. Assuming the temperature gradient in the solvent is rapidly established, we predict that the polymer will respond to it and collapse or not according to our theoretical considerations. We should emphasize that because our predictions suggest a large change in the size of the polymer the effects are not subtle. Consequently, we envision that our predictions are amenable to test by standard light scattering experiments.

(iii) It should

be emphasized that direct measurements of Casimir forces in a variety of situations has not been straightforward. Here, by focusing on the consequences of the GCF on the size of homopolymers we predict large changes in the polymer size. For example, if N ≈ 30,000 then the predicted change in Rg at ∇TC is 531 nm for polystyrene (Rg goes from 825.51 nm to 294.45 nm). This is a large effect and if validated would indirectly provide a measure of GCF, which to date has not been achieved. Although we have focused on polystyrene in neopentane by way of illustrating our theory it should be emphasized that one could consider other solvents and polymers for which similar measurements can be made. We note that a polymer in liquid mixtures can also be profoundly affected by concentration fluctuations. In this case, a temperature gradient induces long-range concentration fluctuations through the Soret effect. We hope to report the consequences of the Soret effect on the size of a polymer elsewhere.

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Finally, it is worth noting that polymers in a binary mixture of good solvents near a critical point exhibit unexpected behavior. 41–43 In this situation, the polymer size is determined by the proximity to the critical point of the mixture. The solvent density fluctuations can induce a partial chain collapse even in a mixture of good solvents, a prediction that has been tested by computer simulations 44–49 and in experimental studies. 50–54 A number of theoretical studies have been devoted to describe coil-globule transitions in dilute polymer solutions caused by various external stimuli. 55–60 In all these examples the short range fluctuation-induced forces arise at equilibrium, and have little bearing to the phenomenon explored here.

Acknowledgement This work was supported by the National Science Foundation under Grant Nos. DMR1401449 and CHE-16-36424. DT acknowledges additional support from the Collie-Welch Regents Chair (F-0019).

Supporting Information In the Supporting Information we provide details on the calculation of the effect of the GCF on a polymer in toluene or in neopentane. In addition, we discuss an alternative way to obtain the scaling law derived in the main text.

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Giant Casimir Non-Equilibrium Forces Drive Coil to Globule Transition

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