SUPPORTING INFORMATION. Line Roughness. in Lamellae-Forming Block Copolymer Films

Transcription

1 SUPPORTING INFORMATION. Line Roughness in Lamellae-Forming Bloc Copolymer Films Ricardo Ruiz *, Lei Wan, Rene Lopez, Thomas R. Albrecht HGST, a Western Digital Company, San Jose, CA 9535, United States Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 7599, United States. I. The correlation coefficient in reciprocal space, C. In the text, e pointed that the expressions in eq. 3 and eq. 5 in the main text form a parallel set in real and reciprocal space: Real Space Reciprocal Space S) G f ) G f ) C G f ) c S4) c p + + p 4 S) S3) G p f ) G f ) + C G f ) S5) f ) G G f ) + G 4 p f ) S6) Where c is related to the covariance of the to edges in a line:

2 ), cov c S7) And the covariance is given by: N ) ) ), cov S8) And C is a corresponding correlation coefficient in reciprocal space. In this section, e ill demonstrate that the parallel relations hold true at every frequency and e ill also demonstrate that C is the Cosine of the phase difference beteen the Fourier components of the opposite edges at frequency f. Eq S and eq S are straightforard to derive substituting the idth and placement definitions eq in the main text) into the definition of variance eq in the main text). We illustrate here the example of eq S. According to eq and in the main text: ) ) ) ) ), cov ) ) ) ) ) ) ) ) ) N N N N N S9) Where in the last line of eq S9 e substituted eq S8 in the last term. Then e substitute the linear correlation coefficient from eq S7 into the last line of eq S9 to obtain:

3 + cov, ) + c S0) Recalling that for self-similar lines in the limit of large N:, then e obtain the familiar form of eq S: S) c No, to obtain eq S3, e first rerite eq S: c S) No e rerite eq S: e c S3) p Equating eq S to S3 and solving for : + p 4 S4) Which is the same as eq S3. No e proceed to do a similar exercise in reciprocal space. First e start ith the poer spectral density PSD) of the idth roughness, G. From the definition of the PSD explained in eq 4 of the main text: 3

4 ;, S5) Where W are the Fourier coefficients given by: W e iπ / N S6) Recall from the definition of eq in the main text:, W W e ) e iπ / N iπ / N e iπ / N S7) Note that each term in eq S7 corresponds to the Fourier coefficients of each line edge, thus: W E E S8) Note the parallel form of eq S8) ith the definition of eq in the main text:. Next e substitute eq S8 into eq S5 in hat follos, to simplify the notation, e restrict the expressions to the range,. We ill leave the special cases of 0 and N/ as an exercise to the interested reader): E E S9) E + E E E S0) Note that the first to terms in eq S0 are the PSD of each line edge:, therefore, eq S0 can be ritten as: 4

5 + E + E S) Note the parallel relationship of eq S ith the last line of eq S9. If e tae the last term in eq S to be the parallel representation of the covariance, it is therefore natural to propose the parallel correlation coefficient in reciprocal space: S) Substituting eq S into eq S, + E + E S3) Recalling that for self-similar lines in the limit of large N, E E E and, then eq S3 becomes E S4) And using the definition of the PSD: and substituting into eq S4: S5) Which demonstrates eq S4. We leave it as an exercise for the interested reader to sho the corresponding relationships for the PSD of the line placement G p : First, it can be shon that in analogy to eq in the main text: P E + E S6) 5

6 Then one can prove eq S5. Next, combining eq S4 and S5 one obtains eq S6. Experimentally, once G and G p are computed, one can obtain C ithout eeping trac of the complex Fourier coefficients Ε or Ε. By solving for G in eq S4 and substituting into eq S5, one arrives at the convenient expression for C : S7) Which is hat e used in the main text eq 6 naturally, there is a parallel expression in real space: ). No, to understand the meaning of C, e proceed to simplify eq S. Since every Fourier coefficient is a complex number, let s define: E E E E e iα e iα S8) Where α and α are the phase values of the Fourier components of the first and second line edges at frequency. Then e substitute eq S8 into eq S: S9) Recalling again that for self-similar lines in the limit of large N: E E E and thus eq S9 can be simplified as: S30) 6

7 Cos S3) Therefore, C represents the Cosine of the phase difference beteen the Fourier components of opposite edges at frequency. In the same ay as its real space analog c), C ranges from to -. When the to Fourier components of opposite edges are perfectly in phase α α ), they are fully correlated and C. When α - α π, they are fully anti-correlated and C -. It is interesting to note that hile the phase information of each E is lost in the PSDs, the phase difference is still preserved in C and can be recovered from the PSDs through eq S7. II Thermal fluctuation in a D membrane. In this section e derive the thermal fluctuation modes for a D interface to sho that the expression differs from the D version only by a constant given by /t, here t is the film thicness. We follo the treatment shon by Safran. To simplify notation, e calculate here the thermal fluctuations of an interface interfacial or capillary modes only). Extension to the membrane is straight forard by adding the curvature term for undulations and curvature and volume terms for peristaltic modes. 3 7

8 Consider the surface of Figure S. The equilibrium flat surface is set to be parallel to the x-y plane. The height of the interface is represented as z hx,y). or z h here is the position vector. The dimension of the surface is L along the x-axis and t along the y-axis in our experiments, t is the thicness of the film). We represent the partial derivatives of h as: h ; h S3) In the Monge gauge, the Area of the surface is: +h +h + h + h S33) Given an interfacial energy, γ, the free energy of the interface is given by: + h + h S34) Which can be separated in to terms: + h + h + S35) Where the first term is a constant given by the size of the interface on the x-y plane F o γlt). The second term is the excess free energy arising from the surface roughness. Thus, h + h S36) No e proceed to calculate the thermal fluctuations of the interface for the one-dimensional case. Note that by one-dimensional e mean that the interface has no fluctuations along the y- direction: h 0, ust lie in the schematic of Fig. S This interpretation of D is needed 8

9 because γ is still a surface term. We no follo the treatment shon in chapter 3.3 of the textboo by Safran but ith the condition that h h and h y 0. Let s start by reriting the excess free energy of eq S36 for the D case: h h S37) We also use the same Fourier pairs as Safran, but noting that they are no D: h h h S38) h Where L is the length of the interface along the x-axis and q is the D ave vector. Note that h has units of [length] hile h has units of [length] 3/. No our goal is to express eq S37 in reciprocal space. First, e calculate h x using eq S38: h h h h S39) No e calculate h x ) : h h h h h h S40) No substituting bac into eq S37: 9

10 h h h h S4) Note that +, so the integral is equal to L hen q -q and it is zero otherise. Thus, e can rerite eq S4: h h S4) Folloing the thermodynamic arguments explained by Safran, if the Hamiltonian the energy of the system) is of the form: h h S43) Then by virtue of the equipartition theorem, the poer spectrum of the fluctuations is given by: h S44) Where B is the Boltzmann constant. The corresponding average in real space is given by: h h S45) Where, according to Safran, d is the dimensionality of the system. We point out that eq S44 is the poer spectral density of the fluctuations as presented in the main text. Similarly, eq S45 corresponds to the variance. Going bac to our particular example, by comparing eq S4 to eq S43, e see that for our problem, the function gq) is given by: 0

11 S46) Therefore the PSD of the fluctuations from eq S44 is given by: h h / S47) Comparing eq S47 ith the D result by Safran, e see that the only difference is the factor /t that as a consequence of the fact that h y 0. Other than that, the functional form is the same ith the same available modes. The exercise done here can be extended to include the energy from the curvature of a membrane see Safran) to obtain the expression for the undulatory modes G und f ) as shon in the main text eq 0 main text), but the reader can quicly identify the /t term that has the same origin as in this simpler example. Finally note that the PSD units in the D case eq S47) are [length] 3 hile in the corresponding D case see Safran) they ould be [length] 4. AUTHOR INFORMATION Corresponding Author *ricardo.ruiz@hgst.com Present Addresses Molecular Vista, Inc., San Jose, CA 959 REFERENCES

12 . Bunday, B. D.; Bishop, M.; McCormac Jr., D. W.; Villarrubia, J. S.; Vladar, A. E.; Dixson, R.; Vorburger, T. V.; Ori, N. G.; Allgair, J. A., Determination of Optimal Parameters for Cd-Sem Measurement of Line-Edge Roughness. Proc. SPIE 004, Safran, S. A., Statistical Thermodynamics of Surfaces, Interfaces, and Membranes. Addison-Wesley Pub.: Huang, H. W., Deformation Free Energy of Bilayer Membrane and Its Effect on Gramicidin Channel Lifetime. Biophys. J. 986, 50, 06.